The Subtraction and Addition Properties of Equality
Solve Equations That Require Simplification
In the previous examples, we were able to isolate the variable with just one operation. Most of the equations we encounter in algebra will take more steps to solve. Usually, we will need to simplify one or both sides of an equation before using the Subtraction or Addition Properties of Equality.
You should always simplify as much as possible before you try to isolate the variable. Remember that to simplify an expression means to do all the operations in the expression. Simplify one side of the equation at a time. Note that simplification is different from the process used to solve an equation in which we apply an operation to both sides.
Example 2.6
How to Solve Equations That Require Simplification
Solve: \(9x−5−8x−6=7\).
Solution
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Step 1. Simplify the expressions on each side as much as possible. |
Rearrange the terms, using the Commutative Property of Addition. Combine like terms. Notice that each side is now simplified as much as possible. |
\(9 x-5-8 x-6=7\) \(9 x-8 x-5-6=7\) \(x-11=7\) |
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Step 2. Isolate the variable. |
Now isolate \(x\). Undo subtraction by adding 11 to both sides. |
\(x-11+11=7+11\) |
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Step 3. Simplify the expressions on both sides of the equation. |
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\(x=18\) |
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Step 4. Check the solution. |
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Check: Substitute \(x=18\). \( \begin{align} 9 x-5-8 x-6 &=7 \\ 9(18)-5-8(18)-6 & \stackrel{?}{=} 7 \\ 162-5-144-6 &\stackrel{?}{=} 7 \\ 157-144-6 & \stackrel{?}{=} 7 \\ 13-6 &\stackrel{?}{=} 7 \\ 7 &=7 \text{✓} \end{align} \) The solution to \(9 x-5-8 x-6=7\) is \(x=18\) |
Try It 2.11
Solve: \(8y−4−7y−7=4\).
Try It 2.12
Solve: \(6z+5−5z−4=3\).
Example 2.7
Solve: \(5(n−4)−4n=−8\).
Solution
We simplify both sides of the equation as much as possible before we try to isolate the variable.
| \(5(n-4)-4n=-8\) | |
| Distribute on the left. | \(5n-20-4n=-8\) |
| Use the Commutative Property to rearrange terms. | \(5n-4n-20=-8\) |
| Combine like terms. | \(n-20=-8\) |
| Each side is as simplified as possible. Next, isolate n. | |
| Undo subtraction by using the Addition Property of Equality. | \(n-20+20=-8+20\) |
| Add. | \(n = 12\) |
| Check. Substitute n=12. |
\(5(n-4)-4n=-8\) \(5(12-4)-4(12) \stackrel{?}{=} -8z\) \(5(8) -48 \stackrel{?}{=} -8\) \(40-48 \stackrel{?}{=} -8\) \(-8 = -8\) ✓ |
| The solution to \(5(n−4)−4n=−8\) is \(n=12\). |
Try It 2.13
Solve: \(5(p−3)−4p=−10\).
Try It 2.14
Solve: \(4(q+2)−3q=−8\).
Example 2.8
Solve: \(3(2y−1)−5y=2(y+1)−2(y+3)\).
Solution
We simplify both sides of the equation before we isolate the variable.
| \(3(2y-1)-5y = 2(y+1) -2(y+3)\) |
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| Distribute on both sides. | \(6y -3- 5y = 2y +2 -2y -6\) |
| Use the Commutative Property of Addition. | \(6y - 5y - 3 = 2y -2y +2 -6\) |
| Combine like terms. | \(y - 3 = -4\) |
| Each side is as simplified as possible. Next, isolate \(y\). | |
| Undo subtraction by using the Addition Property of Equality. | \(y - 3 +3 = -4 +3\) |
| Add. | \(y = -1\) |
| Check. Let \(y=−1\). |
\(3(2y-1)-5=2(y+1)-2(y+3)\) \(3(2(-1)-1)-5(-1)\stackrel{?}{=}2(-1+1)-2(-1+3)\) \(3(-2-1)+5 \stackrel{?}{=} 2(0)-2(2)\) \(3(-3)+5 \stackrel{?}{=} -4\) \(-9+5\stackrel{?}{=}-4\) \(-4 = -4\) ✓ |
| The solution to \(3(2y−1)−5y=2(y+1)−2(y+3)\) is \(y=−1\). |
Try It 2.15
Solve: \(4(2h−3)−7h=6(h−2)−6(h−1)\).
Try It 2.16
Solve: \(2(5x+2)−9x=3(x−2)−3(x−4)\).