8.3 Solve Equations with Variables and Constants on Both Sides

Solving Equations with Variables and Constants on Both Sides

When you have an equation like $5x + 3 = 2x + 15$, variables and constants appear on both sides. Your job is to rearrange everything so the variable ends up alone on one side. This section builds directly on the one- and two-step equations you've already learned, just with an extra move or two.

Equations with Constants on Both Sides (Quick Review)

Before tackling variables on both sides, make sure you're solid on the basics. To isolate a variable, you undo whatever operation is attached to it by doing the opposite operation to both sides.

• If a constant is added to the variable, subtract it from both sides: $x + 3 = 7$ becomes $x = 4$
• If a constant is subtracted from the variable, add it to both sides: $x - 4 = 2$ becomes $x = 6$
• If a constant is multiplying the variable, divide both sides: $3x = 15$ becomes $x = 5$
• If a constant is dividing the variable, multiply both sides: $\frac{x}{2} = 6$ becomes $x = 12$

The key idea: whatever you do to one side, you must do to the other. That keeps the equation balanced.

Equations with Variables on Both Sides

This is the main skill for section 8.3. When variables appear on both sides, you need to move them all to one side first, then deal with the constants.

Step-by-step process:

1. Move the variables to one side. Pick whichever side you prefer (choosing the side where the variable term is larger keeps things positive). Use addition or subtraction to eliminate the variable from the other side.
2. Move the constants to the opposite side. Add or subtract to get all the plain numbers away from the variable.
3. Isolate the variable. Divide or multiply to get the variable completely alone.

Example: Solve $5x + 3 = 2x + 15$

1. Subtract $2x$ from both sides: $3x + 3 = 15$
2. Subtract $3$ from both sides: $3x = 12$
3. Divide both sides by $3$: $x = 4$

Check your answer by plugging it back in: $5(4) + 3 = 23$ and $2(4) + 15 = 23$. Both sides match, so $x = 4$ is correct.

Strategy for More Complex Equations

Some equations look messy at first because there are multiple variable terms and constants scattered on each side. Here's how to handle them:

1. Simplify each side separately by combining like terms.
2. Move all variable terms to one side (add or subtract).
3. Move all constants to the other side (add or subtract).
4. Isolate the variable (divide or multiply).

Example: Solve $3x - 2 + 4x = 2x + 3 + x - 1$

1. Combine like terms on each side: $7x - 2 = 3x + 2$

2. Subtract $3x$ from both sides: $4x - 2 = 2$

3. Add $2$ to both sides: $4x = 4$

4. Divide both sides by $4$: $x = 1$

Special Cases: No Solution and Infinite Solutions

Not every equation has exactly one answer. Watch for these two situations:

No solution happens when the variables cancel out and you're left with a false statement.

• Example: $2x + 5 = 2x - 3$ → subtract $2x$ from both sides → $5 = -3$
• That's never true, so there is no value of $x$ that works. The equation has no solution.

Infinite solutions (identity) happens when the variables cancel out and you're left with a true statement.

• Example: $3(x + 2) = 3x + 6$ → distribute → $3x + 6 = 3x + 6$ → subtract $3x$ → $6 = 6$
• That's always true, so every value of $x$ works. The equation is called an identity.

Combining Like Terms

Before you can solve, you often need to simplify by combining like terms, which are terms with the same variable raised to the same power.

• $3x$ and $-x$ are like terms (both have $x$). Combine them: $3x - x = 2x$
• $2y$ and $4y$ are like terms (both have $y$). Combine them: $2y + 4y = 6y$
• $3x$ and $2y$ are not like terms, so they can't be combined.
• Plain numbers (constants) like $7$ and $-2$ are also like terms with each other.

Always combine like terms on each side of the equation before you start moving things across the equals sign. It makes every step after that much cleaner.