maths.free › Algebra › 5. Systems of Linear Equations › Solving Systems of Equations by Substitution

Solving Systems of Equations by Substitution

Solve a system of equations by substitution

Solve a System of Equations by Substitution

We will use the same system we used first for graphing.

\[\{\begin{array}{l}2x+y=7 \\ x-2y=6\end{array}\]

We will first solve one of the equations for either x or y. We can choose either equation and solve for either variable—but we’ll try to make a choice that will keep the work easy.

Then we substitute that expression into the other equation. The result is an equation with just one variable—and we know how to solve those!

After we find the value of one variable, we will substitute that value into one of the original equations and solve for the other variable. Finally, we check our solution and make sure it makes both equations true.

We’ll fill in all these steps now in .

How to Solve a System of Equations by Substitution

Try it.

Solve the system by substitution. \(\{\begin{array}{l}2x+y=7 \\ x-2y=6\end{array}\)

 

 

 

Solution

If one of the equations in the system is given in slope–intercept form, Step 1 is already done! We’ll see this in .

Example

Try it.

Solve the system by substitution.

\(\{\begin{array}{l}x+y=-1 \\ y=x+5\end{array}\)

Solution

The second equation is already solved for y. We will substitute the expression in place of y in the first equation.

The second equation is already solved for y.
We will substitute into the first equation.
Replace the y with x + 5.
Solve the resulting equation for x.
Substitute x = −3 into y = x + 5 to find y.
The ordered pair is (−3, 2).
Check the ordered pair in both equations:

\(\begin{array}{llllllllllllllll}\begin{array}{lll}x+y & = & -1 \\ -3+2 & \overset{?}{=} & -1 \\ -1 & = & -1\ ✓\end{array} & & & \begin{array}{lll}y & = & x+5 \\ 2 & \overset{?}{=} & -3+5 \\ 2 & = & 2\ ✓\end{array}\end{array}\)
The solution is (−3, 2).

Condensed — the full section is in OpenStax Elementary Algebra 2e.

Solve Applications of Systems of Equations by Substitution

We’ll copy here the problem solving strategy we used in the Solving Systems of Equations by Graphing section for solving systems of equations. Now that we know how to solve systems by substitution, that’s what we’ll do in Step 5.

Some people find setting up word problems with two variables easier than setting them up with just one variable. Choosing the variable names is easier when all you need to do is write down two letters. Think about this in the next example—how would you have done it with just one variable?

Example

Try it.

The sum of two numbers is zero. One number is nine less than the other. Find the numbers.

Solution
Step 1. Read the problem.
Step 2. Identify what we are looking for.We are looking for two numbers.
Step 3. Name what we are looking for. Let \(n=\) the first number
Let \(m=\) the second number
Step 4. Translate into a system of equations.The sum of two numbers is zero.
One number is nine less than the other.
The system is:
Step 5. Solve the system of
equations. We will use substitution
since the second equation is solved
for n.
Substitute m − 9 for n in the first equation.
Solve for m.
Substitute \(m=\frac{9}{2}\) into the second equation
and then solve for n.
Step 6. Check the answer in the problem.Do these numbers make sense in
the problem? We will leave this to you!
Step 7. Answer the question.The numbers are \(\frac{9}{2}\) and \(-\frac{9}{2}.\)

In the , we’ll use the formula for the perimeter of a rectangle, P = 2L + 2W.

Condensed — the full section is in OpenStax Elementary Algebra 2e.

Key Concepts

  • Solve a system of equations by substitution
    1. Solve one of the equations for either variable.
    2. Substitute the expression from Step 1 into the other equation.
    3. Solve the resulting equation.
    4. Substitute the solution in Step 3 into one of the original equations to find the other variable.
    5. Write the solution as an ordered pair.
    6. Check that the ordered pair is a solution to both original equations.

Solving Systems of Equations by Substitution

Solve a System of Equations by Substitution

In the following exercises, solve the systems of equations by substitution.

Try it.

\(\{\begin{array}{l}2x+y=-4 \\ 3x-2y=-6\end{array}\)

Solution

\((-2,0)\)

Try it.

\(\{\begin{array}{l}2x+y=-2 \\ 3x-y=7\end{array}\)

Try it.

\(\{\begin{array}{l}x-2y=-5 \\ 2x-3y=-4\end{array}\)

Solution

\((7,6)\)

Try it.

\(\{\begin{array}{l}x-3y=-9 \\ 2x+5y=4\end{array}\)

Try it.

\(\{\begin{array}{l}5x-2y=-6 \\ y=3x+3\end{array}\)

Solution

\((0,3)\)

Try it.

\(\{\begin{array}{l}-2x+2y=6 \\ y=-3x+1\end{array}\)

Try it.

\(\{\begin{array}{l}2x+3y=3 \\ y=\text{-}x+3\end{array}\)

Solution

\((6,-3)\)

Try it.

\(\{\begin{array}{l}2x+5y=-14 \\ y=-2x+2\end{array}\)

Try it.

\(\{\begin{array}{l}2x+5y=1 \\ y=\frac{1}{3}x-2\end{array}\)

Solution

\((3,-1)\)

Try it.

\(\{\begin{array}{l}3x+4y=1 \\ y=-\frac{2}{5}x+2\end{array}\)

Try it.

\(\{\begin{array}{l}3x-2y=6 \\ y=\frac{2}{3}x+2\end{array}\)

Solution

\((6,6)\)

Try it.

\(\{\begin{array}{l}-3x-5y=3 \\ y=\frac{1}{2}x-5\end{array}\)

Try it.

\(\{\begin{array}{l}2x+y=10 \\ -x+y=-5\end{array}\)

Solution

\((5,0)\)

Try it.

\(\{\begin{array}{l}-2x+y=10 \\ -x+2y=16\end{array}\)

Try it.

\(\{\begin{array}{l}3x+y=1 \\ -4x+y=15\end{array}\)

Solution

\((-2,7)\)

Try it.

\(\{\begin{array}{l}x+y=0 \\ 2x+3y=-4\end{array}\)

Try it.

\(\{\begin{array}{l}x+3y=1 \\ 3x+5y=-5\end{array}\)

Solution

\((-5,2)\)

Try it.

\(\{\begin{array}{l}x+2y=-1 \\ 2x+3y=1\end{array}\)

Try it.

\(\{\begin{array}{l}2x+y=5 \\ x-2y=-15\end{array}\)

Solution

\((-1,7)\)

Try it.

\(\{\begin{array}{l}4x+y=10 \\ x-2y=-20\end{array}\)

Try it.

\(\{\begin{array}{l}y=-2x-1 \\ y=-\frac{1}{3}x+4\end{array}\)

Solution

\((-3,5)\)

Try it.

\(\{\begin{array}{l}y=x-6 \\ y=-\frac{3}{2}x+4\end{array}\)

Try it.

\(\{\begin{array}{l}y=2x-8 \\ y=\frac{3}{5}x+6\end{array}\)

Solution

(10, 12)

Try it.

\(\{\begin{array}{l}y=\text{-}x-1 \\ y=x+7\end{array}\)

Try it.

\(\{\begin{array}{l}4x+2y=8 \\ 8x-y=1\end{array}\)

Solution

\((\frac{1}{2},3)\)

Try it.

\(\{\begin{array}{l}-x-12y=-1 \\ 2x-8y=-6\end{array}\)

Try it.

\(\{\begin{array}{l}15x+2y=6 \\ -5x+2y=-4\end{array}\)

Solution

\((\frac{1}{2},-\frac{3}{4})\)

Try it.

\(\{\begin{array}{l}2x-15y=7 \\ 12x+2y=-4\end{array}\)

Try it.

\(\{\begin{array}{l}y=3x \\ 6x-2y=0\end{array}\)

Solution

Infinitely many solutions

Try it.

\(\{\begin{array}{l}x=2y \\ 4x-8y=0\end{array}\)

Try it.

\(\{\begin{array}{l}2x+16y=8 \\ -x-8y=-4\end{array}\)

Solution

Infinitely many solutions

Try it.

\(\{\begin{array}{l}15x+4y=6 \\ -30x-8y=-12\end{array}\)

Try it.

\(\{\begin{array}{l}y=-4x \\ 4x+y=1\end{array}\)

Solution

No solution

Try it.

\(\{\begin{array}{l}y=-\frac{1}{4}x \\ x+4y=8\end{array}\)

Try it.

\(\{\begin{array}{l}y=\frac{7}{8}x+4 \\ -7x+8y=6\end{array}\)

Solution

No solution

Try it.

\(\{\begin{array}{l}y=-\frac{2}{3}x+5 \\ 2x+3y=11\end{array}\)

Solve Applications of Systems of Equations by Substitution

In the following exercises, translate to a system of equations and solve.

Try it.

The sum of two numbers is 15. One number is 3 less than the other. Find the numbers.

Solution

The numbers are 6 and 9.

Try it.

The sum of two numbers is 30. One number is 4 less than the other. Find the numbers.

Try it.

The sum of two numbers is −26. One number is 12 less than the other. Find the numbers.

Solution

The numbers are −7 and −19.

Try it.

The perimeter of a rectangle is 50. The length is 5 more than the width. Find the length and width.

Try it.

The perimeter of a rectangle is 60. The length is 10 more than the width. Find the length and width.

Solution

The length is 20 and the width is 10.

Try it.

The perimeter of a rectangle is 58. The length is 5 more than three times the width. Find the length and width.

Try it.

The perimeter of a rectangle is 84. The length is 10 more than three times the width. Find the length and width.

Solution

The length is 34 and the width is 8.

Try it.

The measure of one of the small angles of a right triangle is 14 more than 3 times the measure of the other small angle. Find the measure of both angles.

Try it.

The measure of one of the small angles of a right triangle is 26 more than 3 times the measure of the other small angle. Find the measure of both angles.

Solution

The measures are 16° and 74°.

Try it.

The measure of one of the small angles of a right triangle is 15 less than twice the measure of the other small angle. Find the measure of both angles.

Try it.

The measure of one of the small angles of a right triangle is 45 less than twice the measure of the other small angle. Find the measure of both angles.

Solution

The measures are 45° and 45°.

Try it.

Maxim has been offered positions by two car dealers. The first company pays a salary of $10,000 plus a commission of $1,000 for each car sold. The second pays a salary of $20,000 plus a commission of $500 for each car sold. How many cars would need to be sold to make the total pay the same?

Try it.

Jackie has been offered positions by two cable companies. The first company pays a salary of $ 14,000 plus a commission of $100 for each cable package sold. The second pays a salary of $20,000 plus a commission of $25 for each cable package sold. How many cable packages would need to be sold to make the total pay the same?

Solution

80 cable packages would need to be sold.

Try it.

Amara currently sells televisions for company A at a salary of $17,000 plus a $100 commission for each television she sells. Company B offers her a position with a salary of $29,000 plus a $20 commission for each television she sells. How many televisions would Amara need to sell for the options to be equal?

Try it.

Mitchell currently sells stoves for company A at a salary of $12,000 plus a $150 commission for each stove he sells. Company B offers him a position with a salary of $24,000 plus a $50 commission for each stove he sells. How many stoves would Mitchell need to sell for the options to be equal?

Solution

Mitchell would need to sell 120 stoves.

Condensed — the full section is in OpenStax Elementary Algebra 2e.

Practice (40)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

  1. Simplify \(-5(3-x)\).
    If you missed this problem, review .

    Jawaabta muuji

    \(-15+5x\)

  2. Simplify \(4-2(n+5)\).
    If you missed this problem, review .

    Jawaabta muuji

    \(-2n-6\)

  3. Solve for \(y\): \(8y-8=32-2y\)
    If you missed this problem, review .

    Jawaabta muuji

    \(y=4\)

  4. Solve for \(x\): \(3x-9y=-3\)
    If you missed this problem, review .

    Jawaabta muuji

    \(x=3y-1\)

  5. Solve the system by substitution. \(\{\begin{array}{l}2x+y=7 \\ x-2y=6\end{array}\)

     

     

     

  6. Solve the system by substitution. \(\{\begin{array}{l}-2x+y=-11 \\ x+3y=9\end{array}\)

    Jawaabta muuji

    \((6,1)\)

  7. Solve the system by substitution. \(\{\begin{array}{l}x+3y=10 \\ 4x+y=18\end{array}\)

    Jawaabta muuji

    \((4,2)\)

  8. Solve the system by substitution.

    \(\{\begin{array}{l}x+y=-1 \\ y=x+5\end{array}\)

    Jawaabta muuji

    The second equation is already solved for y. We will substitute the expression in place of y in the first equation.

    The second equation is already solved for y.
    We will substitute into the first equation.
    Replace the y with x + 5.
    Solve the resulting equation for x.
    Substitute x = −3 into y = x + 5 to find y.
    The ordered pair is (−3, 2).
    Check the ordered pair in both equations:

    \(\begin{array}{llllllllllllllll}\begin{array}{lll}x+y & = & -1 \\ -3+2 & \overset{?}{=} & -1 \\ -1 & = & -1\ ✓\end{array} & & & \begin{array}{lll}y & = & x+5 \\ 2 & \overset{?}{=} & -3+5 \\ 2 & = & 2\ ✓\end{array}\end{array}\)
    The solution is (−3, 2).

  9. Solve the system by substitution. \(\{\begin{array}{l}x+y=6 \\ y=3x-2\end{array}\)

    Jawaabta muuji

    \((2,4)\)

  10. Solve the system by substitution. \(\{\begin{array}{l}2x-y=1 \\ y=-3x-6\end{array}\)

    Jawaabta muuji

    \((-1,-3)\)

  11. Solve the system by substitution. \(\{\begin{array}{l}3x+y=5 \\ 2x+4y=-10\end{array}\)

    Jawaabta muuji

    We need to solve one equation for one variable. Then we will substitute that expression into the other equation.

    Solve for y.

    Substitute into the other equation.
    Replace the y with −3x + 5.
    Solve the resulting equation for x.

    Substitute x = 3 into 3x + y = 5 to find y.

    The ordered pair is (3, −4).
    Check the ordered pair in both equations:

    \(\begin{array}{llllllllllllllllllll}\begin{array}{lll}3x+y & = & 5 \\ 3\cdot 3+(-4) & \overset{?}{=} & 5 \\ 9-4 & \overset{?}{=} & 5 \\ 5 & = & 5\ ✓\end{array} & & & \begin{array}{lll}2x+4y & = & -10 \\ 2\cdot 3+4(-4) & = & -10 \\ 6-16 & \overset{?}{=} & -10 \\ -10 & = & -10\ ✓\end{array}\end{array}\)
    The solution is (3, −4).

  12. Solve the system by substitution. \(\{\begin{array}{l}4x+y=2 \\ 3x+2y=-1\end{array}\)

    Jawaabta muuji

    \((1,-2)\)

  13. Solve the system by substitution. \(\{\begin{array}{l}-x+y=4 \\ 4x-y=2\end{array}\)

    Jawaabta muuji

    \((2,6)\)

  14. Solve the system by substitution. \(\{\begin{array}{l}x-2y=-2 \\ 3x+2y=34\end{array}\)

    Jawaabta muuji

    We will solve the first equation for \(x\) and then substitute the expression into the second equation.

    Solve for x.

    Substitute into the other equation.
    Replace the x with 2y − 2.
    Solve the resulting equation for y.

    Substitute y = 5 into x − 2y = −2 to find x.




    The ordered pair is (8, 5).
    Check the ordered pair in both equations:

    \(\begin{array}{llllllllllllllllllll}\begin{array}{lll}x-2y & = & -2 \\ 8-2\cdot 5 & \overset{?}{=} & -2 \\ 8-10 & \overset{?}{=} & -2 \\ -2 & = & -2\ ✓\end{array} & & & \begin{array}{lll}3x+2y & = & 34 \\ 3\cdot 8+2\cdot 5 & \overset{?}{=} & 34 \\ 24+10 & \overset{?}{=} & 34 \\ 34 & = & 34\ ✓\end{array}\end{array}\)
    The solution is (8, 5).

  15. Solve the system by substitution. \(\{\begin{array}{l}x-5y=13 \\ 4x-3y=1\end{array}\)

    Jawaabta muuji

    \((-2,-3)\)

  16. Solve the system by substitution. \(\{\begin{array}{l}x-6y=-6 \\ 2x-4y=4\end{array}\)

    Jawaabta muuji

    \((6,2)\)

  17. Solve the system by substitution. \(\{\begin{array}{l}y=-2x+5 \\ y=\frac{1}{2}x\end{array}\)

    Jawaabta muuji

    Since both equations are solved for y, we can substitute one into the other.

    Substitute \(\frac{1}{2}x\) for y in the first equation.
    Replace the y with \(\frac{1}{2}x.\)
    Solve the resulting equation. Start
    by clearing the fraction.
    Solve for x.
    Substitute x = 2 into y = \(\frac{1}{2}x\) to find y.

    The ordered pair is (2,1).
    Check the ordered pair in both equations:

    \(\begin{array}{llllllllllllllllll}\begin{array}{lll}y & = & \frac{1}{2}x \\ 1 & \overset{?}{=} & \frac{1}{2}\cdot 2 \\ 1 & = & 1\ ✓\end{array} & & & \begin{array}{lll}y & = & -2x+5 \\ 1 & \overset{?}{=} & -2\cdot 2+5 \\ 1 & = & -4+5 \\ 1 & = & 1\ ✓\end{array}\end{array}\)
    The solution is (2,1).

  18. Solve the system by substitution. \(\{\begin{array}{l}y=3x-16 \\ y=\frac{1}{3}x\end{array}\)

    Jawaabta muuji

    \((6,2)\)

  19. Solve the system by substitution. \(\{\begin{array}{l}y=\text{-}x+10 \\ y=\frac{1}{4}x\end{array}\)

    Jawaabta muuji

    \((8,2)\)

  20. Solve the system by substitution. \(\{\begin{array}{l}4x+2y=4 \\ 6x-y=8\end{array}\)

    Jawaabta muuji

    We need to solve one equation for one variable. We will solve the first equation for y.

    Solve the first equation for y.
    Substitute −2x + 2 for y in the second equation.
    Replace the y with −2x + 2.
    Solve the equation for x.



    Substitute \(x=\frac{5}{4}\) into 4x + 2y = 4 to find y.



    The ordered pair is \((\frac{5}{4},-\frac{1}{2}).\)
    Check the ordered pair in both equations.

    \(\begin{array}{llllllllllllllllllllll}\begin{array}{lll}4x+2y & = & 4 \\ 4(\frac{5}{4})+2(-\frac{1}{2}) & \overset{?}{=} & 4 \\ 5-1 & \overset{?}{=} & 4 \\ 4 & = & 4\ ✓ \\ \\ \\ \\ \\ \end{array} & & & \begin{array}{lll}6x-y & = & 8 \\ 6(\frac{5}{4})-(-\frac{1}{2}) & \overset{?}{=} & 8 \\ \frac{15}{4}-(-\frac{1}{2}) & \overset{?}{=} & 8 \\ \frac{16}{2} & \overset{?}{=} & 8 \\ 8 & = & 8\ ✓\end{array}\end{array}\)
    The solution is \((\frac{5}{4},-\frac{1}{2}).\)

  21. Solve the system by substitution. \(\{\begin{array}{l}x-4y=-4 \\ -3x+4y=0\end{array}\)

    Jawaabta muuji

    \((2,\frac{3}{2})\)

  22. Solve the system by substitution. \(\{\begin{array}{l}4x-y=0 \\ 2x-3y=5\end{array}\)

    Jawaabta muuji

    \((-\frac{1}{2},-2)\)

  23. Solve the system by substitution. \(\{\begin{array}{l}4x-3y=6 \\ 15y-20x=-30\end{array}\)

    Jawaabta muuji

    We need to solve one equation for one variable. We will solve the first equation for x.

    Solve the first equation for x.
    Substitute \(\frac{3}{4}y+\frac{3}{2}\) for x in the second equation.
    Replace the x with \(\frac{3}{4}y+\frac{3}{2}.\)
    Solve for y.

    Since 0 = 0 is a true statement, the system is consistent. The equations are dependent. The graphs of these two equations would give the same line. The system has infinitely many solutions.

  24. Solve the system by substitution. \(\{\begin{array}{l}2x-3y=12 \\ -12y+8x=48\end{array}\)

    Jawaabta muuji

    infinitely many solutions

  25. Solve the system by substitution. \(\{\begin{array}{l}5x+2y=12 \\ -4y-10x=-24\end{array}\)

    Jawaabta muuji

    infinitely many solutions

  26. Solve the system by substitution. \(\{\begin{array}{l}5x-2y=-10 \\ y=\frac{5}{2}x\end{array}\)

    Jawaabta muuji

    The second equation is already solved for y, so we can substitute for y in the first equation.

    Substitute x for y in the first equation.
    Replace the y with \(\frac{5}{2}x.\)
    Solve for x.

    Since 0 = −10 is a false statement the equations are inconsistent. The graphs of the two equation would be parallel lines. The system has no solutions.

  27. Solve the system by substitution. \(\{\begin{array}{l}3x+2y=9 \\ y=-\frac{3}{2}x+1\end{array}\)

    Jawaabta muuji

    no solution

  28. Solve the system by substitution. \(\{\begin{array}{l}5x-3y=2 \\ y=\frac{5}{3}x-4\end{array}\)

    Jawaabta muuji

    no solution

  29. The sum of two numbers is zero. One number is nine less than the other. Find the numbers.

    Jawaabta muuji
    Step 1. Read the problem.
    Step 2. Identify what we are looking for.We are looking for two numbers.
    Step 3. Name what we are looking for. Let \(n=\) the first number
    Let \(m=\) the second number
    Step 4. Translate into a system of equations.The sum of two numbers is zero.
    One number is nine less than the other.
    The system is:
    Step 5. Solve the system of
    equations. We will use substitution
    since the second equation is solved
    for n.
    Substitute m − 9 for n in the first equation.
    Solve for m.
    Substitute \(m=\frac{9}{2}\) into the second equation
    and then solve for n.
    Step 6. Check the answer in the problem.Do these numbers make sense in
    the problem? We will leave this to you!
    Step 7. Answer the question.The numbers are \(\frac{9}{2}\) and \(-\frac{9}{2}.\)

  30. The sum of two numbers is 10. One number is 4 less than the other. Find the numbers.

    Jawaabta muuji

    The numbers are 3 and 7.

  31. The sum of two number is −6. One number is 10 less than the other. Find the numbers.

    Jawaabta muuji

    The numbers are 2 and −8.

  32. The perimeter of a rectangle is 88. The length is five more than twice the width. Find the length and the width.

    Jawaabta muuji
    Step 1. Read the problem.
    Step 2. Identify what you are looking for.We are looking for the length and width.
    Step 3. Name what we are looking for.Let \(L=\) the length
      \(W=\) the width
    Step 4. Translate into a system of equations.The perimeter of a rectangle is 88.
        2L + 2W = P
    The length is five more than twice the width.
    The system is:
    Step 5. Solve the system of equations.
    We will use substitution since the second
    equation is solved for L.

    Substitute 2W + 5 for L in the first equation.
    Solve for W.
    Substitute W = 13 into the second
    equation and then solve for L.
    Step 6. Check the answer in the problem.Does a rectangle with length 31 and width
    13 have perimeter 88? Yes.
    Step 7. Answer the equation.The length is 31 and the width is 13.

  33. The perimeter of a rectangle is 40. The length is 4 more than the width. Find the length and width of the rectangle.

    Jawaabta muuji

    The length is 12 and the width is 8.

  34. The perimeter of a rectangle is 58. The length is 5 more than three times the width. Find the length and width of the rectangle.

    Jawaabta muuji

    The length is 23 and the width is 6.

  35. The measure of one of the small angles of a right triangle is ten more than three times the measure of the other small angle. Find the measures of both angles.

    Jawaabta muuji

    We will draw and label a figure.

    Step 1. Read the problem.
    Step 2. Identify what you are looking for.We are looking for the measures of the angles.
    Step 3. Name what we are looking for.Let \(a=\) the measure of the 1st angle
    \(\ b=\) the measure of the 2nd angle
    Step 4. Translate into a system of equations.The measure of one of the small angles
    of a right triangle is ten more than three
    times the measure of the other small angle.
    The sum of the measures of the angles of
    a triangle is 180.
    The system is:
    Step 5. Solve the system of equations.
    We will use substitution since the first
    equation is solved for a.
    Substitute 3b + 10 for a in the
    second equation.
    Solve for b.
    Substitute b = 20 into the first
    equation and then solve for a.
    Step 6. Check the answer in the problem.We will leave this to you!
    Step 7. Answer the question.The measures of the small angles are
    20 and 70.

  36. The measure of one of the small angles of a right triangle is 2 more than 3 times the measure of the other small angle. Find the measure of both angles.

    Jawaabta muuji

    The measure of the angles are 22 degrees and 68 degrees.

  37. The measure of one of the small angles of a right triangle is 18 less than twice the measure of the other small angle. Find the measure of both angles.

    Jawaabta muuji

    The measure of the angles are 36 degrees and 54 degrees.

  38. Heather has been offered two options for her salary as a trainer at the gym. Option A would pay her $25,000 plus $15 for each training session. Option B would pay her $10,000 + $40 for each training session. How many training sessions would make the salary options equal?

    Jawaabta muuji
    Step 1. Read the problem.
    Step 2. Identify what you are looking for.We are looking for the number of training sessions
    that would make the pay equal.
    Step 3. Name what we are looking for.Let \(s=\) Heather’s salary.
    \(\ n=\) the number of training sessions
    Step 4. Translate into a system of equations.Option A would pay her $25,000 plus $15
    for each training session.
    Option B would pay her $10,000 + $40
    for each training session
    The system is:
    Step 5. Solve the system of equations.
    We will use substitution.
    Substitute 25,000 + 15n for s in the second equation.
    Solve for n.
    Step 6. Check the answer.Are 600 training sessions a year reasonable?
    Are the two options equal when n = 600?
    Step 7. Answer the question.The salary options would be equal for 600 training sessions.

  39. Geraldine has been offered positions by two insurance companies. The first company pays a salary of $12,000 plus a commission of $100 for each policy sold. The second pays a salary of $20,000 plus a commission of $50 for each policy sold. How many policies would need to be sold to make the total pay the same?

    Jawaabta muuji

    There would need to be 160 policies sold to make the total pay the same.

  40. Kenneth currently sells suits for company A at a salary of $22,000 plus a $10 commission for each suit sold. Company B offers him a position with a salary of $28,000 plus a $4 commission for each suit sold. How many suits would Kenneth need to sell for the options to be equal?

    Jawaabta muuji

    Kenneth would need to sell 1,000 suits.

Symbols used here

\pm
plus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.
\neq
not equal
The two sides are different.
\leq,\ \geq
less/greater than or equal
Inequalities that allow equality; < and > exclude it.
\sqrt{x},\ \sqrt[n]{x}
square root, n-th root
The non-negative number whose square (n-th power) is x.
|x|
absolute value / modulus
Distance from zero: |−3| = 3. For a complex number, distance from the origin.
i
imaginary unit
i² = −1.
\log_b x,\ \ln x
logarithm, natural log
The exponent b must be raised to for x; ln uses base e.
\mathbb{N},\ \mathbb{Z},\ \mathbb{Q},\ \mathbb{R},\ \mathbb{C}
number sets
Naturals, integers, rationals, reals, complex numbers.

How to: Solving Systems of Equations by Substitution

  1. Solve a system of equations by substitution
  2. Solve applications of systems of equations by substitution
  3. Solve one of the equations for either variable.
  4. Substitute the expression from Step 1 into the other equation.
  5. Solve the resulting equation.
  6. Substitute the solution in Step 3 into one of the original equations to find the other variable.
  7. Write the solution as an ordered pair.
  8. Check that the ordered pair is a solution to

Questions people ask

What does it mean to solve an equation?

To find every value of the unknown that makes both sides equal. Each step is an operation applied to both sides that keeps the solution set the same, until the unknown stands alone.

Why do I sometimes get two answers?

A quadratic can cross the axis twice, so it can have two solutions. A degree-n polynomial has up to n. The graph shows where each one comes from.

How do I know whether to factor or use the quadratic formula?

Try factoring for a few seconds: look for two numbers that multiply to a·c and add to b. If nothing obvious appears, the discriminant b² − 4ac tells you how many real roots there are, and the formula finds them without guessing.

Ku day inaad ku

Parts of this page are adapted from OpenStax Elementary Algebra 2e (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

In ka badan Algebra

Linear equationsQuadratic equationsSystems of equationsInequalitiesFactoringExpandingSimplifying expressionsFunctions and graphsExponential and logarithmic equationsPolynomial equationsAbsolute value