Equations with Variables on Both Sides - Pre Algebra

1. Solve $4(x - 1) = 2x + 6$.

   $x = 5$. Distribute to get $4x - 4 = 2x + 6$, so $2x = 10$.

2. Solve $7x - 4 = 3x + 16$.

   $x = 5$. Subtract $3x$ and add 4: $4x = 20$.

3. How many solutions does $2(x + 3) = 2x + 6$ have?

   Infinitely many. Both sides are equal for every $x$; simplifying leaves $6 = 6$.

4. Solve $0.4x + 1.5 = 0.1x - 0.3$.

   $x = -6$. Subtract $0.1x$ and 1.5: $0.3x = -1.8$.

5. What is the goal when moving terms in an equation with variables on both sides?

   To collect all variable terms on one side and all constants on the other, using the same inverse operation on both sides.

6. What result while solving shows that a linear equation has no solution?

   A false statement with no variable left, such as $4 = 7$.

7. Solve $-3x + 4 = x - 8$.

   $x = 3$. Add $3x$ and add 8: $12 = 4x$.

8. Solve $5x + 3 = 2x + 12$.

   $x = 3$. Subtract $2x$ and 3 from both sides: $3x = 9$.

9. How many solutions does $3x + 2 = 3x + 5$ have?

   None. Subtracting $3x$ leaves $2 = 5$, which is false.

10. For what value of $k$ does $kx + 2 = 3x + 7$ have no solution?

    $k = 3$. Then the $x$ terms cancel and leave $2 = 7$, which is false.

11. Solve $\frac{1}{2}x + 3 = x - 1$.

    $x = 8$. Subtract $\frac{1}{2}x$ and add 1: $4 = \frac{1}{2}x$.

12. Solve $2x + 9 = 5x - 3$.

    $x = 4$. Subtract $2x$ and add 3: $12 = 3x$.

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