2.3: Solve Equations with Variables and Constants on Both Sides (2024)

Example \(\PageIndex{1}\)

Solve: \(7x+8=−13\).

Solution

In this equation, the variable is found only on the left side. It makes sense to call the left side the “variable” side. Therefore, the right side will be the “constant” side. We will write the labels above the equation to help us remember what goes where.

Since the left side is the “xx”, or variable side, the 8 is out of place. We must “undo” adding 8 by subtracting 8, and to keep the equality we must subtract 8 from both sides.

Use the Subtraction Property of Equality.
Simplify.
Now all the variables are on the left and the constant on the right. The equation looks like those you learned to solve earlier.
Use the Division Property of Equality.
Simplify.
Check:
Let x=−3.

Try It \(\PageIndex{2}\)

Solve: \(3x+4=−8\).

Answer

\(x=−4\)

Try It \(\PageIndex{3}\)

Solve: \(5a+3=−37\).

Answer

\(a=−8\)

Example \(\PageIndex{4}\)

Solve: \(8y−9=31\).

Solution

Notice, the variable is only on the left side of the equation, so we will call this side the “variable” side, and the right side will be the “constant” side. Since the left side is the “variable” side, the 9 is out of place. It is subtracted from the 8y, so to “undo” subtraction, add 9 to both sides. Remember, whatever you do to the left, you must do to the right.

Add 9 to both sides.
Simplify.
		The variables are now on one side and the constants on the other. We continue from here as we did earlier.
Divide both sides by 8.
Simplify.
Check:
Let y=5.

Try It \(\PageIndex{5}\)

Solve: \(5y−9=16\).

Answer

\(y=5\)

Try It \(\PageIndex{6}\)

Solve: \(3m−8=19\).

Answer

\(m = 9\)

Example \(\PageIndex{7}\)

Solve: \(9x=8x−6\).

Solution

Here the variable is on both sides, but the constants only appear on the right side, so let’s make the right side the “constant” side. Then the left side will be the “variable” side.

We don’t want any x’s on the right, so subtract the 8x from both sides.
Simplify.
We succeeded in getting the variables on one side and the constants on the other, and have obtained the solution.
Check:
Let x=−6.

Try It \(\PageIndex{8}\)

Solve: \(6n=5n−10\).

Answer

\(n = -10\)

Try It \(\PageIndex{9}\)

Solve: \(-6c = -7c - 1\)

Answer

\(c = -1\)

Example \(\PageIndex{10}\)

Solve: \(5y - 9 = 8y\)

Solution

The only constant is on the left and the y’s are on both sides. Let’s leave the constant on the left and get the variables to the right.

Subtract 5y from both sides.
Simplify.
We have the y’s on the right and the constants on the left. Divide both sides by 3.
Simplify.
Check:
Let \(y=−3\).

Try It \(\PageIndex{11}\)

Solve: \(3p−14=5p\).

Answer

\(p = -7\)

Try It \(\PageIndex{12}\)

Solve: \(8m + 9 = 5m\)

Answer

\(m = -3\)

Example \(\PageIndex{13}\)

Solve: \(12x = -x + 26\)

Solution

The only constant is on the right, so let the left side be the “variable” side.

Remove the −x from the right side by adding x to both sides.
Simplify.
All the x’s are on the left and the constants are on the right. Divide both sides by 13.
Simplify.

Try It \(\PageIndex{14}\)

Solve: \(12j = -4j + 32\)

Answer

\(j = 2\)

Try It \(\PageIndex{15}\)

Solve: \(8h = -4h + 12\)

Answer

\(h = 1\)

Example \(\PageIndex{16}\): How to Solve Equations with Variables and Constants on Both Sides

Solve: \(7x + 5 = 6x + 2\)

Solution

Try It \(\PageIndex{17}\)

Solve: \(12x+8=6x+2\).

Answer

\(x=−1\)

Try It \(\PageIndex{18}\)

Solve: \(9y+4=7y+12\).

Answer

\(y=4\)

BEGINNING STRATEGY FOR SOLVING EQUATIONS WITH VARIABLES AND CONSTANTS ON BOTH SIDES OF THE EQUATION.

1. Choose which side will be the “variable” side—the other side will be the “constant” side.
2. Collect the variable terms to the “variable” side of the equation, using the Addition or Subtraction Property of Equality.
3. Collect all the constants to the other side of the equation, using the Addition or Subtraction Property of Equality.
4. Make the coefficient of the variable equal 1, using the Multiplication or Division Property of Equality.
5. Check the solution by substituting it into the original equation.

Example \(\PageIndex{19}\)

Solve: \(8n−4=−2n+6\).

Solution

In the first step, choose the variable side by comparing the coefficients of the variables on each side.

Since \(8>−2\), make the left side the “variable” side.
We don’t want variable terms on the right side—add 2n to both sides to leave only constants on the right.
Combine like terms.
We don’t want any constants on the left side, so add 4 to both sides.
Simplify.
The variable term is on the left and the constant term is on the right. To get the coefficient of nn to be one, divide both sides by 10.
Simplify.
Check:
Let n=1.

Try It \(\PageIndex{20}\)

Solve: \(8q - 5 = -4q + 7\)

Answer

\(q = 1\)

Try It \(\PageIndex{21}\)

Solve: \(7n - 3 = n + 3\)

Answer

\(n = 1\)

Example \(\PageIndex{22}\)

Solve: \(7a -3 = 13a + 7\)

Solution

In the first step, choose the variable side by comparing the coefficients of the variables on each side.

Since 13>7, make the right side the “variable” side and the left side the “constant” side.

Subtract 7a from both sides to remove the variable term from the left.
Combine like terms.
Subtract 7 from both sides to remove the constant from the right.
Simplify.
Divide both sides by 6 to make 1 the coefficient of a.
Simplify.
Check:
Let \(a=−\frac{5}{3}\).

Try It \(\PageIndex{23}\)

Solve: \(2a - 2 = 6a + 18\)

Answer

\(a = -5\)

Try It \(\PageIndex{24}\)

Solve: \(4k -1 = 7k + 17\)

Answer

\(k = -6\)

Example \(\PageIndex{25}\)

Solve: \(\frac{4}{5}x + 6 = \frac{1}{4}x - 2\)

Solution

Since \(\frac{5}{4} > \frac{1}{4}\), make the left side the “variable” side and the right side the “constant” side.

Subtract \(\frac{1}{4}x\) from both sides.
Combine like terms.
Subtract 6 from both sides.
Simplify.
Check: Let \(x = -8\)	\(\begin{array} {ccc} {\frac{5}{4}x + 6} &{=} &{\frac{1}{4}x - 2} \\ {\frac{5}{4}(-8) + 6} &{\stackrel{?}{=}} &{\frac{1}{4}(-8) - 2} \\ {-10 + 6} &{\stackrel{?}{=}} &{-2 - 2} \\ {-4} &{=} &{-4\checkmark} \end{array}\)

Try It \(\PageIndex{26}\)

Solve: \(\frac{7}{8}x - 12 = -\frac{1}{8}x - 2\)

Answer

\(x = 10\)

Try It \(\PageIndex{27}\)

Solve: \(\frac{7}{6}x + 11 = \frac{1}{6}y + 8\)

Answer

\(y = -3\)

Example \(\PageIndex{28}\)

Solve: \(7.8x+4=5.4x−8\).

Solution

Since \(7.8>5.4\), make the left side the “variable” side and the right side the “constant” side.

Subtract 5.4x from both sides.
Combine like terms.
Subtract 4 from both sides.
Simplify.
Use the Division Property of Equality.
Simplify.
Check:
Let \(x=−5\)

Try It \(\PageIndex{29}\)

Solve: \(2.8x + 12 = -1.4x - 9\)

Answer

\(x = -5\)

Try It \(\PageIndex{30}\)

Solve: \(3.6y + 8 = 1.2y - 4\)

Answer

\(y = -5\)