Question

solving linear equations
common core algebra ii homework
fluency
date:

1. solve each of the following linear equations. if the equation is inconsistent, state so. if the equation is an identity, also state so. reduce any non - integer answers to fractions in simplest form.

(a) $7x + 5 = 2x - 35$
(b) $\frac{x}{3}-7=-5$
(c) $4x + 5 = 4x - 1$
(d) $\frac{5(x - 3)}{2}-1 = 14$
(e) $3(x - 1)+2 = x + 9$
(f) $4x-(2x - 1)=x + 5 + x - 6$
(g) $5(2x - 6)+2(4x + 3)=8x - 9$
(h) $\frac{2x + 5}{6}=\frac{x}{18}$ (cross multiply to begin)
(i) $\frac{10x - 4}{2}+7 = 5(x + 1)$
(j) $18 - 2(x + 7)=\frac{8x - 20}{2}-2$
core algebra ii, unit #1 - essential algebra concepts - lesson #2
ction, red hook, ny 12571, © 2015

Explanation:

(a) Step1: Isolate x terms

$7x - 2x = -35 - 5$

(a) Step2: Simplify both sides

$5x = -40$

(a) Step3: Solve for x

$x = \frac{-40}{5} = -8$

(b) Step1: Isolate the fraction term

$\frac{x}{3} = -5 + 7$

(b) Step2: Simplify right side

$\frac{x}{3} = 2$

(b) Step3: Solve for x

$x = 2 \times 3 = 6$

(c) Step1: Isolate x terms

$4x - 4x = -1 - 5$

(c) Step2: Simplify both sides

$0 = -6$

(d) Step1: Isolate the fraction term

$\frac{5(x-3)}{2} = 14 + 1$

(d) Step2: Simplify right side

$\frac{5(x-3)}{2} = 15$

(d) Step3: Eliminate denominator

$5(x-3) = 15 \times 2 = 30$

(d) Step4: Divide by 5

$x - 3 = \frac{30}{5} = 6$

(d) Step5: Solve for x

$x = 6 + 3 = 9$

(e) Step1: Expand left side

$3x - 3 + 2 = x + 9$

(e) Step2: Simplify left side

$3x - 1 = x + 9$

(e) Step3: Isolate x terms

$3x - x = 9 + 1$

(e) Step4: Simplify and solve

$2x = 10 \implies x = 5$

(f) Step1: Expand left side

$4x - 2x + 1 = x + 5 + x - 6$

(f) Step2: Simplify both sides

$2x + 1 = 2x - 1$

(f) Step3: Isolate x terms

$2x - 2x = -1 - 1$

(f) Step4: Simplify

$0 = -2$

(g) Step1: Expand left side

$10x - 30 + 8x + 6 = 8x - 9$

(g) Step2: Simplify left side

$18x - 24 = 8x - 9$

(g) Step3: Isolate x terms

$18x - 8x = -9 + 24$

(g) Step4: Simplify and solve

$10x = 15 \implies x = \frac{15}{10} = \frac{3}{2}$

(h) Step1: Cross multiply

$18(2x + 5) = 6x$

(h) Step2: Expand left side

$36x + 90 = 6x$

(h) Step3: Isolate x terms

$36x - 6x = -90$

(h) Step4: Simplify and solve

$30x = -90 \implies x = -3$

(i) Step1: Simplify the fraction

$\frac{10x-4}{2} = 5x - 2$, so $5x - 2 + 7 = 5(x+1)$

(i) Step2: Simplify left side

$5x + 5 = 5(x+1)$

(i) Step3: Expand right side

$5x + 5 = 5x + 5$

(i) Step4: Isolate x terms

$5x - 5x = 5 - 5$

(i) Step5: Simplify

$0 = 0$

(j) Step1: Expand left side

$18 - 2x - 14 = \frac{8x-20}{2} - 2$

(j) Step2: Simplify left side

$4 - 2x = \frac{8x-20}{2} - 2$

(j) Step3: Simplify right fraction

$\frac{8x-20}{2} = 4x - 10$, so $4 - 2x = 4x - 10 - 2$

(j) Step4: Simplify right side

$4 - 2x = 4x - 12$

(j) Step5: Isolate x terms

$-2x - 4x = -12 - 4$

(j) Step6: Simplify and solve

$-6x = -16 \implies x = \frac{-16}{-6} = \frac{8}{3}$

Answer:

(a) $x = -8$
(b) $x = 6$
(c) Inconsistent (no solution)
(d) $x = 9$
(e) $x = 5$
(f) Inconsistent (no solution)
(g) $x = \frac{3}{2}$
(h) $x = -3$
(i) Identity (all real numbers are solutions)
(j) $x = \frac{8}{3}$