Linear Equations - GED Math

1. What rule keeps an equation balanced while you solve it?

   Do the same operation to both sides: add, subtract, multiply or divide both sides by the same amount (never divide by zero).

2. Solving an equation leaves a false statement such as $3 = 5$. How many solutions does it have?

   None. The variable dropped out and the sides can never be equal.

3. Solve $5x - 4 = 2x + 11$.

   $x = 5$. Subtract $2x$ and add $4$: $3x = 15$.

4. What is a quick way to clear fractions from an equation?

   Multiply every term on both sides by the least common denominator of the fractions.

5. Solve $-2x > 8$.

   $x < -4$. Dividing by $-2$ reverses the inequality sign.

6. Solve $2(x + 3) = 14$.

   $x = 4$. Divide by $2$ to get $x + 3 = 7$, then subtract $3$.

7. Solve $5x - 8 = 27$.

   $x = 7$. Add $8$ to get $5x = 35$, then divide by $5$.

8. Solve $\frac{x}{3} + 2 = 6$.

   $x = 12$. Subtract $2$ to get $\frac{x}{3} = 4$, then multiply by $3$.

9. Solving an equation leaves a true statement such as $4 = 4$. How many solutions does it have?

   Infinitely many. Both sides are the same expression, so every value of the variable works.

10. Solve $P = 2l + 2w$ for $w$.

    $w = \frac{P - 2l}{2}$. Subtract $2l$ from both sides, then divide by $2$.

11. Solve $0.2x + 1 = 3$.

    $x = 10$. Subtract $1$ to get $0.2x = 2$, then divide by $0.2$.

12. What must you do to an inequality sign when you multiply or divide both sides by a negative number?

    Reverse it. For example, $<$ becomes $>$.

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