Linear Equations

8.B.1Variables both sides01/04

Solve equations with variables on both sides.

Solve for x

5 x - 3 = 2 x + 12

Step01of 04

When a variable appears on BOTH sides, the goal is the same: get all variable terms on one side, all constants on the other.

Strategy. Subtract the SMALLER variable term from both sides (keeps the remaining coefficient positive — fewer sign errors).

8.B.1 — Solve equations with variables on both sides.

Solve for x

5 x - 3 = 2 x + 12

Step 01 of 04

When a variable appears on BOTH sides, the goal is the same: get all variable terms on one side, all constants on the other.

Strategy. Subtract the SMALLER variable term from both sides (keeps the remaining coefficient positive — fewer sign errors).

Step 02 of 04

Step 1 — collect variable terms. Subtract $2 x$ from both sides.

5 x - 2 x - 3 = 2 x - 2 x + 12

3 x - 3 = 12

Step 03 of 04

Step 2 — collect constants. Add 3 to both sides.

3 x = 15

Step 3 — divide.

x = 5

Step 04 of 04

Always check by plugging in.

5 (5) - 3 = 22 vs. 2 (5) + 12 = 22 ✓

Both sides equal — solution confirmed.

Key insight

Move all the variables to one side, all the numbers to the other. Subtract the smaller variable term first to keep coefficients positive. Always verify by substituting back into the original equation.

8.B.2 — Equations with parentheses.

Step 01 of 04

Distribute first. Clear all parentheses before doing anything else. The order is: distribute → combine like terms on each side → collect variables on one side → solve.

3 (2 x + 4) = 4 (x + 5)

Step 02 of 04

Distribute both sides.

6 x + 12 = 4 x + 20

Now it's a standard "variables on both sides" problem.

Step 03 of 04

Solve.

6 x - 4 x = 20 - 12

2 x = 8 \Rightarrow x = 4

Step 04 of 04

Negative-distribution trap. A leading minus is really $- 1$ ; flip every sign inside.

- (2 x - 7) + 5 = 3 x

- 2 x + 7 + 5 = 3 x \Rightarrow 12 = 5 x \Rightarrow x = 12/5

Key insight

Distribute first (every term, every sign). Then collect like terms, then collect variables, then solve. The leading minus flips signs inside the parens — that's the most common error.

8.B.3 — Number of solutions — one, none, or infinite.

Step 01 of 04

When you solve a linear equation, three outcomes are possible.

Single solution	→variable cancels in interesting way; you get $x = c$
No solution	→variable cancels and you get a FALSE statement (e.g. $5 = 7$ )
Infinitely many	→variable cancels and you get a TRUE statement (e.g. $5 = 5$ )

Step 02 of 04

One solution example.

3 x + 5 = 2 x + 11 \Rightarrow x = 6

No solution example.

2 x + 4 = 2 x - 7

4 = - 7 (false) \Rightarrow no solution

The variable terms cancel completely; what's left is a contradictory numeric statement. No $x$ can make it true.

Step 03 of 04

Infinite solutions example.

3 (x + 2) = 3 x + 6

3 x + 6 = 3 x + 6 \Rightarrow 0 = 0 (always true)

Any value of $x$ satisfies — the equation is an IDENTITY.

Step 04 of 04

Geometric reading. Each side is a line.

Different slopes	→lines cross once → ONE solution
Same slope, different y-intercept	→parallel lines → NO solution
Same slope, same y-intercept	→same line → INFINITELY MANY solutions

Key insight

If the variable cancels: TRUE statement → all real numbers work; FALSE statement → nothing works. Otherwise, exactly one solution. Same logic as looking at two lines: cross once, parallel forever, or coincide.

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8.B.1 – 8.B.3 · CCSS 8.EE.C.7 · Full curriculum →