Question

solving multi-step equations
student handout
name
date
pd

a. label each part of the equation at the right.

b. explain what it means to solve an equation:

solving equations
steps to solve

• the goal in solving an equation is to __________ the variable by using __________, or opposite, operations.

1. ____________ if necessary
2. ____________ any like terms.
3. isolate the variable:

a: use addition/subtraction to remove the ____________.
b: use multiplication/division to remove the ____________.
explain in your own words how you can check your solution after solving:

solve each of the equations below. be sure to show all work and your check - step.

$8x - 14 + 16x + 5 = 39$
check:

$-3(12x + 9) = 99$
check:

$-(2x + 5) + 14 = 22$
check:

$3.4(x + 7) - 2.1x = 35.5$
check:

Explanation:

Part a: Label each part of the equation \( 30x + 8 = -6 \)

• \( 30x \): Term with variable (coefficient \( 30 \), variable \( x \))
• \( + \): Operation (addition)
• \( 8 \): Constant term
• \( = \): Equality sign
• \( -6 \): Constant term on the right side

Part b: Explain what it means to solve an equation

The goal in solving an equation is to isolate the variable by using inverse, or opposite, operations.

Steps to Solve:

1. Simplify if necessary (combine like terms, distribute, etc.)
2. Combine any like terms.
3. Isolate the variable:

a. Use addition/subtraction to remove the constant term (from the side with the variable).
b. Use multiplication/division to remove the coefficient (of the variable).

Checking a Solution:

To check a solution, substitute the value of the variable back into the original equation. Simplify both sides. If both sides are equal, the solution is correct.

Solve \( 8x - 14 + 16x + 5 = 39 \)

Step 1: Combine like terms

\( 8x + 16x - 14 + 5 = 39 \)
\( 24x - 9 = 39 \)

Step 2: Isolate the variable (add 9 to both sides)

\( 24x - 9 + 9 = 39 + 9 \)
\( 24x = 48 \)

Step 3: Divide by 24

\( \frac{24x}{24} = \frac{48}{24} \)
\( x = 2 \)

Check:

Substitute \( x = 2 \) into \( 8x - 14 + 16x + 5 \):
\( 8(2) - 14 + 16(2) + 5 = 16 - 14 + 32 + 5 = 2 + 32 + 5 = 39 \)
Left side = Right side (\( 39 = 39 \)), so \( x = 2 \) is correct.

Solve \( -3(12x + 6) = 99 \)

Step 1: Distribute \( -3 \)

\( -3(12x) + (-3)(6) = 99 \)
\( -36x - 18 = 99 \)

Step 2: Add 18 to both sides

\( -36x - 18 + 18 = 99 + 18 \)
\( -36x = 117 \)

Step 3: Divide by \( -36 \)

\( x = \frac{117}{-36} = -\frac{13}{4} = -3.25 \)

Check:

Substitute \( x = -3.25 \) into \( -3(12x + 6) \):
\( -3(12(-3.25) + 6) = -3(-39 + 6) = -3(-33) = 99 \)
Left side = Right side (\( 99 = 99 \)), so \( x = -3.25 \) is correct.

Solve \( -(2x + 5) + 14 = 22 \)

Step 1: Distribute the negative sign

\( -2x - 5 + 14 = 22 \)

Step 2: Combine like terms

\( -2x + 9 = 22 \)

Step 3: Subtract 9 from both sides

\( -2x + 9 - 9 = 22 - 9 \)
\( -2x = 13 \)

Step 4: Divide by \( -2 \)

\( x = -\frac{13}{2} = -6.5 \)

Check:

Substitute \( x = -6.5 \) into \( -(2x + 5) + 14 \):
\( -(2(-6.5) + 5) + 14 = -(-13 + 5) + 14 = -(-8) + 14 = 8 + 14 = 22 \)
Left side = Right side (\( 22 = 22 \)), so \( x = -6.5 \) is correct.

Solve \( 3.4(x + 7) - 2.1x = 35.5 \)

Step 1: Distribute \( 3.4 \)

\( 3.4x + 23.8 - 2.1x = 35.5 \)

Step 2: Combine like terms

\( (3.4x - 2.1x) + 23.8 = 35.5 \)
\( 1.3x + 23.8 = 35.5 \)

Step 3: Subtract 23.8 from both sides

\( 1.3x + 23.8 - 23.8 = 35.5 - 23.8 \)
\( 1.3x = 11.7 \)

Step 4: Divide by \( 1.3 \)

\( x = \frac{11.7}{1.3} = 9 \)

Check:

Substitute \( x = 9 \) into \( 3.4(x + 7) - 2.1x \):
\( 3.4(16) - 2.1(9) = 54.4 - 18.9 = 35.5 \)
Left side = Right side (\( 35.5 = 35.5 \)), so \( x = 9 \) is correct.

Final Answers:

• \( 8x - 14 + 16x + 5 = 39 \): \( \boldsymbol{x = 2} \)
• \( -3(12x + 6) = 99 \): \( \boldsymbol{x = -3.25} \) (or \( -\frac{13}{4} \))
• \( -(2x + 5) + 14 = 22 \): \( \boldsymbol{x = -6.5} \) (or \( -\frac{13}{2} \))
• \( 3.4(x + 7) - 2.1x = 35.5 \): \( \boldsymbol{x = 9} \)

Answer:

Part a: Label each part of the equation \( 30x + 8 = -6 \)

• \( 30x \): Term with variable (coefficient \( 30 \), variable \( x \))
• \( + \): Operation (addition)
• \( 8 \): Constant term
• \( = \): Equality sign
• \( -6 \): Constant term on the right side

Part b: Explain what it means to solve an equation

The goal in solving an equation is to isolate the variable by using inverse, or opposite, operations.

Steps to Solve:

1. Simplify if necessary (combine like terms, distribute, etc.)
2. Combine any like terms.
3. Isolate the variable:

a. Use addition/subtraction to remove the constant term (from the side with the variable).
b. Use multiplication/division to remove the coefficient (of the variable).

Checking a Solution:

To check a solution, substitute the value of the variable back into the original equation. Simplify both sides. If both sides are equal, the solution is correct.

Solve \( 8x - 14 + 16x + 5 = 39 \)

Step 1: Combine like terms

\( 8x + 16x - 14 + 5 = 39 \)
\( 24x - 9 = 39 \)

Step 2: Isolate the variable (add 9 to both sides)

\( 24x - 9 + 9 = 39 + 9 \)
\( 24x = 48 \)

Step 3: Divide by 24

\( \frac{24x}{24} = \frac{48}{24} \)
\( x = 2 \)

Check:

Substitute \( x = 2 \) into \( 8x - 14 + 16x + 5 \):
\( 8(2) - 14 + 16(2) + 5 = 16 - 14 + 32 + 5 = 2 + 32 + 5 = 39 \)
Left side = Right side (\( 39 = 39 \)), so \( x = 2 \) is correct.

Solve \( -3(12x + 6) = 99 \)

Step 1: Distribute \( -3 \)

\( -3(12x) + (-3)(6) = 99 \)
\( -36x - 18 = 99 \)

Step 2: Add 18 to both sides

\( -36x - 18 + 18 = 99 + 18 \)
\( -36x = 117 \)

Step 3: Divide by \( -36 \)

\( x = \frac{117}{-36} = -\frac{13}{4} = -3.25 \)

Check:

Substitute \( x = -3.25 \) into \( -3(12x + 6) \):
\( -3(12(-3.25) + 6) = -3(-39 + 6) = -3(-33) = 99 \)
Left side = Right side (\( 99 = 99 \)), so \( x = -3.25 \) is correct.

Solve \( -(2x + 5) + 14 = 22 \)

Step 1: Distribute the negative sign

\( -2x - 5 + 14 = 22 \)

Step 2: Combine like terms

\( -2x + 9 = 22 \)

Step 3: Subtract 9 from both sides

\( -2x + 9 - 9 = 22 - 9 \)
\( -2x = 13 \)

Step 4: Divide by \( -2 \)

\( x = -\frac{13}{2} = -6.5 \)

Check:

Substitute \( x = -6.5 \) into \( -(2x + 5) + 14 \):
\( -(2(-6.5) + 5) + 14 = -(-13 + 5) + 14 = -(-8) + 14 = 8 + 14 = 22 \)
Left side = Right side (\( 22 = 22 \)), so \( x = -6.5 \) is correct.

Solve \( 3.4(x + 7) - 2.1x = 35.5 \)

Step 1: Distribute \( 3.4 \)

\( 3.4x + 23.8 - 2.1x = 35.5 \)

Step 2: Combine like terms

\( (3.4x - 2.1x) + 23.8 = 35.5 \)
\( 1.3x + 23.8 = 35.5 \)

Step 3: Subtract 23.8 from both sides

\( 1.3x + 23.8 - 23.8 = 35.5 - 23.8 \)
\( 1.3x = 11.7 \)

Step 4: Divide by \( 1.3 \)

\( x = \frac{11.7}{1.3} = 9 \)

Check:

Substitute \( x = 9 \) into \( 3.4(x + 7) - 2.1x \):
\( 3.4(16) - 2.1(9) = 54.4 - 18.9 = 35.5 \)
Left side = Right side (\( 35.5 = 35.5 \)), so \( x = 9 \) is correct.

Final Answers:

• \( 8x - 14 + 16x + 5 = 39 \): \( \boldsymbol{x = 2} \)
• \( -3(12x + 6) = 99 \): \( \boldsymbol{x = -3.25} \) (or \( -\frac{13}{4} \))
• \( -(2x + 5) + 14 = 22 \): \( \boldsymbol{x = -6.5} \) (or \( -\frac{13}{2} \))
• \( 3.4(x + 7) - 2.1x = 35.5 \): \( \boldsymbol{x = 9} \)