Question

1. name the property of equality that was used to justify each step involved in solving the equation in the table below

steps | name of the property that was used
$2(5x - 7) = 2x + 10$ | given
$10x - 14 = 2x + 10$ |
$-14 = -8x + 10$ |
$-24 = -8x$ |
$3 = x$ |
$x = 3$ |

1. solve for the indicated variable

a) $y - y_1 = m(x - x_1)$ for $x$
b) $k = \frac{mv^2}{2}$ for $v$

1. find the lcm for each set of numbers

a) 63, 42, 84
b) 27, 36, 35, 14

Explanation:

Problem 7

Step1: Identify property for expanding

The distributive property (or distributive property of multiplication over subtraction) is used to expand $2(5x-7)$ to $10x-14$.

Step2: Identify property for subtracting $10x$

Subtraction Property of Equality: Subtract $10x$ from both sides of $10x-14=2x+10$ to get $-14=-8x+10$.

Step3: Identify property for subtracting 10

Subtraction Property of Equality: Subtract 10 from both sides of $-14=-8x+10$ to get $-24=-8x$.

Step4: Identify property for dividing by $-8$

Division Property of Equality: Divide both sides of $-24=-8x$ by $-8$ to get $3=x$.

Step5: Identify property for swapping sides

Symmetric Property of Equality: Swap the sides of $3=x$ to get $x=3$.

Part a)

Step1: Isolate the term with $x$

Divide both sides by $m$ (where $m
eq0$)
$\frac{y-y_1}{m} = x - x_1$

Step2: Solve for $x$

Add $x_1$ to both sides
$x = \frac{y-y_1}{m} + x_1$

Part b)

Step1: Eliminate the denominator

Multiply both sides by 2
$2K = mv^2$

Step2: Isolate $v^2$

Divide both sides by $m$ (where $m
eq0$)
$v^2 = \frac{2K}{m}$

Step3: Solve for $v$

Take square root of both sides
$v = \pm\sqrt{\frac{2K}{m}}$

Part a)

Step1: Prime factorize each number

$63 = 3^2 \times 7$
$42 = 2 \times 3 \times 7$
$84 = 2^2 \times 3 \times 7$

Step2: Take highest powers of primes

Multiply the highest power of each prime: $2^2 \times 3^2 \times 7$
$\text{LCM} = 4 \times 9 \times 7 = 252$

Part b)

Step1: Prime factorize each number

$27 = 3^3$
$36 = 2^2 \times 3^2$
$35 = 5 \times 7$
$14 = 2 \times 7$

Step2: Take highest powers of primes

Multiply the highest power of each prime: $2^2 \times 3^3 \times 5 \times 7$
$\text{LCM} = 4 \times 27 \times 5 \times 7 = 3780$

Answer:

Steps	Name of the property that was used
$10x - 14 = 2x + 10$	Distributive Property
$-14 = -8x + 10$	Subtraction Property of Equality
$-24 = -8x$	Subtraction Property of Equality
$3 = x$	Division Property of Equality
$x = 3$	Symmetric Property of Equality

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Problem 8