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Solve Equations Using Integers; The Division Property of Equality

Determine whether an integer is a solution of an equation

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 EqualityAddition 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.

Now you Không máy tính nào có thể tính toán được nó, nhưng các mảnh của nó có thể tính toán được. Thử một trong những cách dưới đây, hoặc gõ theo ý bạn.

Luyện tập (40)

Đầu tiên thử mỗi câu trên giấy. Cho thấy câu trả lời để kiểm tra; những câu trả lời đã kiểm tra có thể mở trong trình giải cho mỗi bước.

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

    Giải đáp

    \(0\)

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

    Giải đáp

    \(16\)

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

    Giải đáp

    \(x-5\)

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

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

    Giải đáp

    ⓐ 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. ⓒ \(\ x=-3\)

    Giải đáp
    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. ⓒ \(\ y=-7\)

    Giải đáp
    1. ⓐ no
    2. ⓑ no
    3. ⓒ yes

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

    Giải đáp
    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:

    \(y+11=7\)

    Giải đáp

    −4

  9. Solve:

    \(y+15=-4\)

    Giải đáp

    −19

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

    Giải đáp
    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:

    \(a-2=-8\)

    Giải đáp

    −6

  12. Solve:

    \(n-4=-8\)

    Giải đáp

    −4

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

    Giải đáp

    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.

    Giải đáp

    4x = 12; x = 3

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

    Giải đáp

    3x = 6; x = 2

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

    Giải đáp

    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:

    \(8a=56\)

    Giải đáp

    7

  18. Solve:

    \(11n=121\)

    Giải đáp

    11

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

    Giải đáp

    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:

    \(-8p=96\)

    Giải đáp

    −12

  21. Solve:

    \(-12m=108\)

    Giải đáp

    −9

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

    Giải đáp
    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 \(-2\).

    Giải đáp

    x + 7 = −2; x = −9

  24. Translate and solve:

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

    Giải đáp

    y + 11 = 2; y = −9

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

    Giải đáp
    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 \(-4\).

    Giải đáp

    p − 2 = −4; p = −2

  27. Translate and solve:

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

    Giải đáp

    q − 7 = −3; q = 4

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

    Giải đáp
    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 \(y\).

    Giải đáp

    132 = −12y; y = −11

  30. Translate and solve:

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

    Giải đáp

    117 = −13z; z = −9

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

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

    Giải đáp
    1. ⓐ no
    2. ⓑ no
    3. ⓒ yes

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

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

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

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

    Giải đáp
    1. ⓐ no
    2. ⓑ no
    3. ⓒ yes

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

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

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

    Giải đáp

    x = −16

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

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

    Giải đáp

    r = −14

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

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

    Giải đáp

    p = 3

  40. \(-120=10q\)

    Giải đáp

    q = −12

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Làm thế nào: 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.

Câu hỏi người ta hỏi

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.

Một số phần của trang này được chuyển từ OpenStax Prealgebra 2e (CC BY-NC-SA 4.0). Được nén lại và giải thích lại ở đây; lỗi là của chúng tôi. Bởi vì nguồn của nó được cấp phép cho sử dụng không thương mại, bài học này là miễn phí, không bán gì, và được chia sẻ dưới cùng một giấy phép.

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