Question

part ii – constructed response
solve each equation.

1. $5 = 4.5 + c$
2. $4 - 2x = 15$
3. $-8 = -2 - 3p$
4. $\frac{y}{4} + 10 = -14$
5. $-3(2a - 3) + 1 = -20$
6. $4w + 2 - w = 7 - 2w$
7. solve each equation below. state whether each equation has one solution, no solutions, or infinitely many solutions.

a. $y + 13 + 7y - 7 = 2(4y + 3)$ number of solutions:
b. $25 - 2z - 10z = -3(4z + 8) - 1$ number of solutions:
c. $3 - 2(x + 1) = 2x - 7$ number of solutions:
unit 1 - equations

Explanation:

Step1: Isolate c via subtraction

$5 - 4.5 = 4.5 + c - 4.5$

Step2: Calculate c value

$0.5 = c$

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Step1: Isolate x term via subtraction

$4 - 4 - 2x = 15 - 4$
$ -2x = 11$

Step2: Solve for x via division

$x = \frac{11}{-2} = -\frac{11}{2}$

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Step1: Isolate p term via addition

$-8 + 2 = -2 + 2 - 3p$
$-6 = -3p$

Step2: Solve for p via division

$p = \frac{-6}{-3} = 2$

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Step1: Isolate $\frac{y}{4}$ via subtraction

$\frac{y}{4} + 10 - 10 = -14 - 10$
$\frac{y}{4} = -24$

Step2: Solve for y via multiplication

$y = -24 \times 4 = -96$

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Step1: Expand the left-hand side

$-3(2a - 3) + 1 = -6a + 9 + 1 = -6a + 10$
$-6a + 10 = -20$

Step2: Isolate a term via subtraction

$-6a + 10 - 10 = -20 - 10$
$-6a = -30$

Step3: Solve for a via division

$a = \frac{-30}{-6} = 5$

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Step1: Combine like terms on both sides

$4w - w + 2 = 7 - 2w$
$3w + 2 = 7 - 2w$

Step2: Isolate w terms via addition

$3w + 2w + 2 = 7$
$5w + 2 = 7$

Step3: Isolate w term via subtraction

$5w = 7 - 2 = 5$

Step4: Solve for w via division

$w = \frac{5}{5} = 1$

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7a.

Step1: Combine like terms on left-hand side

$y + 7y + 13 - 7 = 8y + 6$

Step2: Expand right-hand side

$2(4y + 3) = 8y + 6$

Step3: Compare both sides

$8y + 6 = 8y + 6$
Subtract $8y + 6$ from both sides: $0 = 0$

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7b.

Step1: Combine like terms on left-hand side

$25 - 2z - 10z = 25 - 12z$

Step2: Expand right-hand side

$-3(4z + 8) - 1 = -12z - 24 - 1 = -12z - 25$

Step3: Compare both sides

$25 - 12z = -12z - 25$
Add $12z$ to both sides: $25 = -25$

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7c.

Step1: Expand left-hand side

$3 - 2(x + 1) = 3 - 2x - 2 = 1 - 2x$
$1 - 2x = 2x - 7$

Step2: Isolate x terms via addition

$1 + 7 = 2x + 2x$
$8 = 4x$

Step3: Solve for x via division

$x = \frac{8}{4} = 2$

Answer:

1. $c = 0.5$
2. $x = -\frac{11}{2}$
3. $p = 2$
4. $y = -96$
5. $a = 5$
6. $w = 1$
7. a. Infinitely many solutions

b. No solutions
c. One solution ($x=2$)