Lesson 2.3: Solve Equations with Variables and Constants on Both Sides

New Concept

Ready to level up your equation-solving skills? This lesson introduces a powerful strategy for equations with variables and constants on both sides. We'll learn how to organize any linear equation by isolating variables on one side and constants on the other.

What’s next

Next, you'll see this strategy in action with interactive examples. Then, you'll apply what you've learned on a series of practice cards to build your confidence.

Property

Our strategy will involve choosing one side of the equation to be the “variable side”, and the other side of the equation to be the “constant side.” Then, we will use the Subtraction and Addition Properties of Equality to get all the variable terms together on one side of the equation and the constant terms together on the other side. By doing this, we will transform the equation into the form $ax = b$.

Examples

• Solve $4x + 5 = -15$. First, subtract 5 from both sides to get $4x = -20$. Then, divide both sides by 4 to find $x = -5$.

• Solve $6y - 7 = 23$. First, add 7 to both sides to get $6y = 30$. Then, divide both sides by 6 to find $y = 5$.

Property

For equations with variables on both sides of the equation, begin as we did above—choose a “variable” side and a “constant” side, and then use the subtraction and addition properties of equality to collect all variables on one side and all constants on the other side.

Examples

• Solve $10x = 9x - 7$. Subtract $9x$ from both sides to get all the variables on the left, which gives the solution $x = -7$.

• Solve $3y - 8 = 6y$. Subtract $3y$ from both sides to gather the variables on the right, which gives $-8 = 3y$. Divide by 3 to get $y = -\frac{8}{3}$.

Property

When an equation has variables and constants on both sides, it may take several steps to solve. We need a clear and organized strategy. The strategy is to first collect the variable terms to one side of the equation, and then collect the constant terms to the other side.

Examples

• Solve $8x + 4 = 7x + 9$. First, subtract $7x$ from both sides to get $x + 4 = 9$. Then, subtract 4 from both sides to find $x = 5$.

• Solve $10n - 5 = -3n + 21$. First, add $3n$ to both sides to get $13n - 5 = 21$. Then, add 5 to both sides to get $13n = 26$. Finally, divide by 13 to find $n = 2$.

Property

Step 1. Choose which side will be the “variable” side—the other side will be the “constant” side. A helpful approach is to make the “variable” side the side that has the variable with the larger coefficient.
Step 2. Collect the variable terms to the “variable” side of the equation, using the Addition or Subtraction Property of Equality.
Step 3. Collect all the constants to the other side of the equation.
Step 4. Make the coefficient of the variable equal 1.
Step 5. Check the solution by substituting it into the original equation.

Examples

• Solve $9k - 1 = 12k + 14$. Choose the right side for variables since $12 > 9$. This leads to $-1 - 14 = 12k - 9k$, which simplifies to $-15 = 3k$, so $k = -5$.

• Solve $\frac{3}{2}x + 4 = \frac{1}{2}x - 6$. Choose the left side for variables since $\frac{3}{2} > \frac{1}{2}$. This leads to $x + 4 = -6$, so $x = -10$.