Solve Equations Using Integers; The Division Property of Equality

Determine Whether a Number is a Solution of an Equation

In Solve Equations with the Subtraction and Addition Properties of Equality, we saw that a solution of an equation is a value of a variable that makes a true statement when substituted into that equation. In that section, we found solutions that were whole numbers. Now that we’ve worked with integers, we’ll find integer solutions to equations.

The steps we take to determine whether a number is a solution to an equation are the same whether the solution is a whole number or an integer.

Example

Try it.

Determine whether each of the following is a solution of \(2x-5=-13\text{:}\)

1. ⓐ \(\ x=4\)
2. ⓑ \(\ x=-4\)
3. ⓒ \(\ x=-9.\)

Solution

ⓐ Substitute 4 for x in the equation to determine if it is true.
Multiply.
Subtract.

Since \(x=4\) does not result in a true equation, \(4\) is not a solution to the equation.

ⓑ Substitute −4 for x in the equation to determine if it is true.
Multiply.
Subtract.

Since \(x=-4\) results in a true equation, \(-4\) is a solution to the equation.

ⓒ Substitute −9 for x in the equation to determine if it is true.
Substitute −9 for x.
Multiply.
Subtract.

Since \(x=-9\) does not result in a true equation, \(-9\) is not a solution to the equation.

Solve Equations with Integers Using the Addition and Subtraction Properties of Equality

In Solve Equations with the Subtraction and Addition Properties of Equality, we solved equations similar to the two shown here using the Subtraction and Addition Properties of Equality. Now we can use them again with integers.

When you add or subtract the same quantity from both sides of an equation, you still have equality.

Example

Try it.

Solve: \(y+9=5.\)

Solution

Subtract 9 from each side to undo the addition.
Simplify.

Check the result by substituting \(-4\) into the original equation.

	\(y+9=5\)
Substitute −4 for y	\(-4+9\overset{?}{=}5\)
	\(5=5✓\)

Since \(y=-4\) makes \(y+9=5\) a true statement, we found the solution to this equation.

Example

Try it.

Solve: \(a-6=-8\)

Solution

Add 6 to each side to undo the subtraction.
Simplify.
Check the result by substituting \(-2\) into the original equation:
Substitute \(-2\) for \(a\)

The solution to \(a-6=-8\) is \(-2.\)

Since \(a=-2\) makes \(a-6=-8\) a true statement, we found the solution to this equation.

Model the Division Property of Equality

All of the equations we have solved so far have been of the form \(x+a=b\) or \(x-a=b.\) We were able to isolate the variable by adding or subtracting the constant term. Now we’ll see how to solve equations that involve division.

We will model an equation with envelopes and counters in .

Here, there are two identical envelopes that contain the same number of counters. Remember, the left side of the workspace must equal the right side, but the counters on the left side are “hidden” in the envelopes. So how many counters are in each envelope?

To determine the number, separate the counters on the right side into \(2\) groups of the same size. So \(6\) counters divided into \(2\) groups means there must be \(3\) counters in each group (since \(6\div 2=3).\)

What equation models the situation shown in ? There are two envelopes, and each contains \(x\) counters. Together, the two envelopes must contain a total of \(6\) counters. So the equation that models the situation is \(2x=6.\)

We can divide both sides of the equation by \(2\) as we did with the envelopes and counters.

We found that each envelope contains \(\text{3 counters.}\) Does this check? We know \(2\cdot 3=6,\) so it works. Three counters in each of two envelopes does equal six.

Example

Try it.

Write an equation modeled by the envelopes and counters, and then solve it.

Solution

There are \(\text{4 envelopes,}\) or \(4\) unknown values, on the left that match the \(\text{8 counters}\) on the right. Let’s call the unknown quantity in the envelopes \(x.\)

Write the equation.
Divide both sides by 4.
Simplify.

There are \(\text{2 counters}\) in each envelope.

Condensed: the full section is in OpenStax Prealgebra 2e.

Solve Equations Using the Division Property of Equality

The previous examples lead to the Division Property of Equality. When you divide both sides of an equation by any nonzero number, you still have equality.

Example

Try it.

\(\text{Solve:}\ 7x=-49.\)

Solution

To isolate \(x,\) we need to undo multiplication.

Divide each side by 7.
Simplify.

Check the solution.

	\(7x=-49\)
Substitute −7 for x.	\(7(-7)\overset{?}{=}-49\)
	\(-49=-49✓\)

Therefore, \(-7\) is the solution to the equation.

Example

Try it.

Solve: \(-3y=63.\)

Solution

To isolate \(y,\) we need to undo the multiplication.

Divide each side by −3.
Simplify

Check the solution.

	\(-3y=63\)
Substitute −21 for y.	\(-3(-21)\overset{?}{=}63\)
	\(63=63✓\)

Since this is a true statement, \(y=-21\) is the solution to the equation.

Translate to an Equation and Solve

In the past several examples, we were given an equation containing a variable. In the next few examples, we’ll have to first translate word sentences into equations with variables and then we will solve the equations.

Example

Try it.

Translate and solve: five more than \(x\) is equal to \(-3.\)

Solution

	five more than \(x\) is equal to \(-3\)
Translate	\(x+5=-3\)
Subtract \(5\) from both sides.	\(x+5-5=-3-5\)
Simplify.	\(x=-8\)

Check the answer by substituting it into the original equation.

\(\begin{array}{l} \\ \\ x+5=-3\ \\ -8+5\overset{?}{=}-3\ \\ -3=-3✓\end{array}\)

Example

Try it.

Translate and solve: the difference of \(n\) and \(6\) is \(-10.\)

Solution

	the difference of \(n\) and \(6\) is \(-10\)
Translate.	\(n-6=-10\)
Add \(6\) to each side.	\(n-6+6=-10+6\)
Simplify.	\(n=-4\)

Check the answer by substituting it into the original equation.

\(\begin{array}{l} \\ n-6=-10\ \\ -4-6\overset{?}{=}-10\ \\ -10=-10✓\end{array}\)

Example

Try it.

Translate and solve: the number \(108\) is the product of \(-9\) and \(y.\)

Solution

	the number of \(108\) is the product of \(-9\) and \(y\)
Translate.	\(108=-9y\)
Divide by \(-9\).	\(\frac{108}{-9}=\frac{-9y}{-9}\)
Simplify.	\(-12=y\)

Check the answer by substituting it into the original equation.

\(\begin{array}{l}108=-9y \\ 108\overset{?}{=}-9(-12) \\ 108=108✓\end{array}\)

Key Concepts

• How to determine whether a number is a solution to an equation.
  • Step 1. Substitute the number for the variable in the equation.
  • Step 2. Simplify the expressions on both sides of the equation.
  • Step 3. Determine whether the resulting equation is true.
    • If it is true, the number is a solution.
    • If it is not true, the number is not a solution.
• Properties of Equalities

  Subtraction Property of Equality	Addition Property of Equality
  \(\text{For any numbers}\ a,b,c,\) \(\text{if}\ a=b\ \text{then}\ a-c=b-c.\)	\(\text{For any numbers}\ a,b,c,\) \(\text{if}\ a=b\ \text{then}\ a+c=b+c.\)

• Division Property of Equality
  • For any numbers \(a,b,c,\) and \(c\ne 0\)
    If \(a=b\), then \(\frac{a}{c}=\frac{b}{c}\).

Chapter Practice Test

In the following exercises, compare the numbers, using \(<\ \text{or}\ >\ \text{or}\ =\text{.}\)

In the following exercises, find the opposite of each number.

In the following exercises, simplify.

In the following exercises, evaluate.

In the following exercises, translate each phrase into an algebraic expression and then simplify, if possible.

In the following exercises, solve.

In the following exercises, solve.

Condensed: the full section is in OpenStax Prealgebra 2e.

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Übung (40)

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1. \(\text{Evaluate}\ x+4\ \text{when}\ x=-4.\)

   Schritt für Schritt durcharbeiten Tipp

   Antwort aufdecken

   \(0\)

2. \(\text{Solve:}\ y-6=10.\)

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   Antwort aufdecken

   \(16\)

3. Translate into an algebraic expression \(5\) less than \(x.\)

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   Antwort aufdecken

   \(x-5\)

4. Determine whether each of the following is a solution of \(2x-5=-13\text{:}\)
   

   1. ⓐ \(\ x=4\)
   2. ⓑ \(\ x=-4\)
   3. ⓒ \(\ x=-9.\)

   Schritt für Schritt durcharbeiten Tipp

   Antwort aufdecken

   ⓐ Substitute 4 for x in the equation to determine if it is true.
   Multiply.
   Subtract.

   Since \(x=4\) does not result in a true equation, \(4\) is not a solution to the equation.

   ⓑ Substitute −4 for x in the equation to determine if it is true.
   Multiply.
   Subtract.

   Since \(x=-4\) results in a true equation, \(-4\) is a solution to the equation.

   ⓒ Substitute −9 for x in the equation to determine if it is true.
   Substitute −9 for x.
   Multiply.
   Subtract.

   Since \(x=-9\) does not result in a true equation, \(-9\) is not a solution to the equation.

5. Determine whether each of the following is a solution of \(2x-8=-14\text{:}\)

   1. ⓐ \(\ x=-11\)
   2. ⓑ \(\ x=11\)
   3. ⓒ \(\ x=-3\)

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   Antwort aufdecken

   1. ⓐ no
   2. ⓑ no
   3. ⓒ yes

6. Determine whether each of the following is a solution of \(2y+3=-11\text{:}\)

   1. ⓐ \(\ y=4\)
   2. ⓑ \(\ y=-4\)
   3. ⓒ \(\ y=-7\)

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   Antwort aufdecken

   1. ⓐ no
   2. ⓑ no
   3. ⓒ yes

7. Solve: \(y+9=5.\)

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   Antwort aufdecken

   Subtract 9 from each side to undo the addition.
   Simplify.

   Check the result by substituting \(-4\) into the original equation.

   	\(y+9=5\)
   Substitute −4 for y	\(-4+9\overset{?}{=}5\)
   	\(5=5✓\)

   Since \(y=-4\) makes \(y+9=5\) a true statement, we found the solution to this equation.

8. Solve:

   \(y+11=7\)

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   Antwort aufdecken

   −4

9. Solve:

   \(y+15=-4\)

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   Antwort aufdecken

   −19

10. Solve: \(a-6=-8\)

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    Antwort aufdecken

    Add 6 to each side to undo the subtraction.
    Simplify.
    Check the result by substituting \(-2\) into the original equation:
    Substitute \(-2\) for \(a\)

    The solution to \(a-6=-8\) is \(-2.\)

    Since \(a=-2\) makes \(a-6=-8\) a true statement, we found the solution to this equation.

11. Solve:

    \(a-2=-8\)

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    Antwort aufdecken

    −6

12. Solve:

    \(n-4=-8\)

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    Antwort aufdecken

    −4

13. Write an equation modeled by the envelopes and counters, and then solve it.

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    Antwort aufdecken

    There are \(\text{4 envelopes,}\) or \(4\) unknown values, on the left that match the \(\text{8 counters}\) on the right. Let’s call the unknown quantity in the envelopes \(x.\)
    

    Write the equation.
    Divide both sides by 4.
    Simplify.

    There are \(\text{2 counters}\) in each envelope.

14. Write the equation modeled by the envelopes and counters. Then solve it.

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    Antwort aufdecken

    4x = 12; x = 3

15. Write the equation modeled by the envelopes and counters. Then solve it.

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    Antwort aufdecken

    3x = 6; x = 2

16. \(\text{Solve:}\ 7x=-49.\)

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    Antwort aufdecken

    To isolate \(x,\) we need to undo multiplication.
    

    Divide each side by 7.
    Simplify.

    Check the solution.

    	\(7x=-49\)
    Substitute −7 for x.	\(7(-7)\overset{?}{=}-49\)
    	\(-49=-49✓\)

    Therefore, \(-7\) is the solution to the equation.

17. Solve:

    \(8a=56\)

    Schritt für Schritt durcharbeiten Tipp

    Antwort aufdecken

    7

18. Solve:

    \(11n=121\)

    Schritt für Schritt durcharbeiten Tipp

    Antwort aufdecken

    11

19. Solve: \(-3y=63.\)

    Schritt für Schritt durcharbeiten Tipp

    Antwort aufdecken

    To isolate \(y,\) we need to undo the multiplication.
    

    Divide each side by −3.
    Simplify

    Check the solution.

    	\(-3y=63\)
    Substitute −21 for y.	\(-3(-21)\overset{?}{=}63\)
    	\(63=63✓\)

    Since this is a true statement, \(y=-21\) is the solution to the equation.

20. Solve:

    \(-8p=96\)

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    Antwort aufdecken

    −12

21. Solve:

    \(-12m=108\)

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    Antwort aufdecken

    −9

22. Translate and solve: five more than \(x\) is equal to \(-3.\)

    Schritt für Schritt durcharbeiten Tipp

    Antwort aufdecken

    	five more than \(x\) is equal to \(-3\)
    Translate	\(x+5=-3\)
    Subtract \(5\) from both sides.	\(x+5-5=-3-5\)
    Simplify.	\(x=-8\)

    Check the answer by substituting it into the original equation.

    \(\begin{array}{l} \\ \\ x+5=-3\ \\ -8+5\overset{?}{=}-3\ \\ -3=-3✓\end{array}\)

23. Translate and solve:

    Seven more than \(x\) is equal to \(-2\).

    Schritt für Schritt durcharbeiten Tipp

    Antwort aufdecken

    x + 7 = −2; x = −9

24. Translate and solve:

    \(\text{Eleven more than}\ y\ \text{is equal to 2.}\)

    Schritt für Schritt durcharbeiten Tipp

    Antwort aufdecken

    y + 11 = 2; y = −9

25. Translate and solve: the difference of \(n\) and \(6\) is \(-10.\)

    Schritt für Schritt durcharbeiten Tipp

    Antwort aufdecken

    	the difference of \(n\) and \(6\) is \(-10\)
    Translate.	\(n-6=-10\)
    Add \(6\) to each side.	\(n-6+6=-10+6\)
    Simplify.	\(n=-4\)

    Check the answer by substituting it into the original equation.

    \(\begin{array}{l} \\ n-6=-10\ \\ -4-6\overset{?}{=}-10\ \\ -10=-10✓\end{array}\)

26. Translate and solve:

    The difference of \(p\) and \(2\) is \(-4\).

    Schritt für Schritt durcharbeiten Tipp

    Antwort aufdecken

    p − 2 = −4; p = −2

27. Translate and solve:

    The difference of \(q\) and \(7\) is \(-3\).

    Schritt für Schritt durcharbeiten Tipp

    Antwort aufdecken

    q − 7 = −3; q = 4

28. Translate and solve: the number \(108\) is the product of \(-9\) and \(y.\)

    Schritt für Schritt durcharbeiten Tipp

    Antwort aufdecken

    	the number of \(108\) is the product of \(-9\) and \(y\)
    Translate.	\(108=-9y\)
    Divide by \(-9\).	\(\frac{108}{-9}=\frac{-9y}{-9}\)
    Simplify.	\(-12=y\)

    Check the answer by substituting it into the original equation.

    \(\begin{array}{l}108=-9y \\ 108\overset{?}{=}-9(-12) \\ 108=108✓\end{array}\)

29. Translate and solve:

    The number \(132\) is the product of \(-12\) and \(y\).

    Schritt für Schritt durcharbeiten Tipp

    Antwort aufdecken

    132 = −12y; y = −11

30. Translate and solve:

    The number \(117\) is the product of \(-13\) and \(z\).

    Schritt für Schritt durcharbeiten Tipp

    Antwort aufdecken

    117 = −13z; z = −9

31. \(4x-2=6\)

    1. ⓐ \(\ x=-2\)
    2. ⓑ \(\ x=-1\)
    3. ⓒ \(\ x=2\)

    Schritt für Schritt durcharbeiten Tipp

    Antwort aufdecken

    1. ⓐ no
    2. ⓑ no
    3. ⓒ yes

32. \(4y-10=-14\)

    1. ⓐ \(\ y=-6\)
    2. ⓑ \(\ y=-1\)
    3. ⓒ \(\ y=1\)

    Schritt für Schritt durcharbeiten Tipp

33. \(9a+27=-63\)

    1. ⓐ \(\ a=6\)
    2. ⓑ \(\ a=-6\)
    3. ⓒ \(\ a=-10\)

    Schritt für Schritt durcharbeiten Tipp

    Antwort aufdecken

    1. ⓐ no
    2. ⓑ no
    3. ⓒ yes

34. \(7c+42=-56\)

    1. ⓐ \(c=2\)
    2. ⓑ \(c=-2\)
    3. ⓒ \(c=-14\)

    Schritt für Schritt durcharbeiten Tipp

35. \(x+(-2)=-18\)

    Schritt für Schritt durcharbeiten Tipp

    Antwort aufdecken

    x = −16

36. \(y+(-3)=-10\)

    Schritt für Schritt durcharbeiten Tipp

37. \(r-(-5)=-9\)

    Schritt für Schritt durcharbeiten Tipp

    Antwort aufdecken

    r = −14

38. \(s-(-2)=-11\)

    Schritt für Schritt durcharbeiten Tipp

39. \(-14p=-42\)

    Schritt für Schritt durcharbeiten Tipp

    Antwort aufdecken

    p = 3

40. \(-120=10q\)

    Schritt für Schritt durcharbeiten Tipp

    Antwort aufdecken

    q = −12

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Tippen Sie auf jedes Symbol für die vollständige Definition, ein Bild und was jeder Buchstabe darin bedeutet.

\neqnot equal
The two sides are different.\pmplus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.\leqless/greater than or equal
Inequalities that allow equality; < and > exclude it.\approxapproximately equal
Equal to the precision shown, not exactly.\sqrt{x}square root, n-th root
The non-negative number whose square (n-th power) is x.a \bmod nremainder
What is left after dividing a by n.\%per cent
Per hundred: 15% = 15/100.a : bratio, fraction
a for every b; a divided by b.

Wie man: Solve Equations Using Integers; The Division Property of Equality

1. Determine whether an integer is a solution of an equation
2. Solve equations with integers using the Addition and Subtraction Properties of Equality
3. Model the Division Property of Equality
4. Solve equations using the Division Property of Equality
5. Translate to an equation and solve
6. Substitute the number for the variable in the equation.
7. Simplify the expressions on both sides of the equation.
8. Determine whether the resulting equation is true.

Fragen, die die Leute stellen

Why does the order of operations matter?

Because 2 + 3 × 4 would otherwise be two different numbers. The convention (brackets, exponents, multiplication and division, addition and subtraction) exists so every reader gets the same value from the same expression.

How do I check an arithmetic answer?

Estimate first (round every number and compute roughly), then compare. If the estimate and the exact answer disagree by more than a little, one of them is wrong. The solver shows every operation, so you can find which line went astray.

Why are fractions harder than decimals?

They are not harder, they are more exact: 1/3 is a precise number, 0.333 is an approximation. Fractions need a common denominator to add, which is the one extra step people trip on.

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