Question

name: solve, then state whether it is one solution, no solution or infinite solutions a. (6m - 2 = m + 13) b. (4y + 9 = 4y - 7) c. (3c + 2 = 3c + 2) d. (3(x - 4) = 2x + 6) e. (-8y + 14 = -2(4y - 7)) f. (18x - 5 = 3(6x - 2)) g. (-8a + 10 = 2(5 - 4a)) h. (9x + 3x - 10 = 3(2x + x)) i. (4x - 10 = x + 3x - 2x) j. (\frac{2}{3}(6x + 3) = 4x + 2) k. (a - 6 = 8 - (9 + a)) l. (3(n - 1) = 5n + 3 - 2n) m. (6(h - 1) = 6h + 4 + 2h) n. (3(2y + 3) = 6y + 9) o. (\frac{7}{8}w = \frac{1}{2}w + \frac{3}{4}w)

Explanation:

A.

Step1: Isolate $m$ terms

$6m - m = 13 + 2$

Step2: Simplify to solve for $m$

$5m = 15 \implies m = \frac{15}{5} = 3$

Step3: Classify solution type

One unique solution.

B.

Step1: Expand and simplify both sides

$4y + 9 = 4y - 7$

Step2: Subtract $4y$ from both sides

$9 = -7$

Step3: Classify solution type

No solution (contradiction).

C.

Step1: Subtract $3x$ from both sides

$2 = 2$

Step2: Classify solution type

Infinite solutions (identity).

D.

Step1: Expand left-hand side

$3x - 12 = 2x + 6$

Step2: Isolate $x$ terms

$3x - 2x = 6 + 12$

Step3: Simplify to solve for $x$

$x = 18$

Step4: Classify solution type

One unique solution.

E.

Step1: Simplify both sides

$-8j + 14 = -8j + 14$

Step2: Classify solution type

Infinite solutions (identity).

F.

Step1: Expand right-hand side

$18x - 5 = 18x - 6$

Step2: Subtract $18x$ from both sides

$-5 = -6$

Step3: Classify solution type

No solution (contradiction).

G.

Step1: Expand right-hand side

$-8a + 10 = 10 - 8a$

Step2: Classify solution type

Infinite solutions (identity).

H.

Step1: Simplify both sides

$12x - 10 = 12x$

Step2: Subtract $12x$ from both sides

$-10 = 0$

Step3: Classify solution type

No solution (contradiction).

I.

Step1: Simplify both sides

$4x - 10 = 2x$

Step2: Isolate $x$ terms

$4x - 2x = 10$

Step3: Simplify to solve for $x$

$2x = 10 \implies x = 5$

Step4: Classify solution type

One unique solution.

J.

Step1: Expand left-hand side

$\frac{2}{3}(6x) + \frac{2}{3}(3) = 4x + 2 \implies 4x + 2 = 4x + 2$

Step2: Classify solution type

Infinite solutions (identity).

K.

Step1: Simplify right-hand side

$a - 6 = 8 - 9 - a \implies a - 6 = -1 - a$

Step2: Isolate $a$ terms

$a + a = -1 + 6$

Step3: Simplify to solve for $a$

$2a = 5 \implies a = \frac{5}{2}$

Step4: Classify solution type

One unique solution.

L.

Step1: Simplify both sides

$3n - 3 = 3n + 3$

Step2: Subtract $3n$ from both sides

$-3 = 3$

Step3: Classify solution type

No solution (contradiction).

M.

Step1: Simplify both sides

$8h - 8 = 8h + 4$

Step2: Subtract $8h$ from both sides

$-8 = 4$

Step3: Classify solution type

No solution (contradiction).

N.

Step1: Expand left-hand side

$6y + 9 = 6y + 9$

Step2: Classify solution type

Infinite solutions (identity).

O.

Step1: Multiply all terms by 8 to eliminate fractions

$7w = 4w + 6w$

Step2: Simplify right-hand side

$7w = 10w$

Step3: Isolate $w$ terms

$7w - 10w = 0 \implies -3w = 0$

Step4: Solve for $w$

$w = 0$

Step5: Classify solution type

One unique solution.

Answer:

A. $m=3$, One solution
B. No solution
C. Infinite solutions
D. $x=18$, One solution
E. Infinite solutions
F. No solution
G. Infinite solutions
H. No solution
I. $x=5$, One solution
J. Infinite solutions
K. $a=\frac{5}{2}$, One solution
L. No solution
M. No solution
N. Infinite solutions
O. $w=0$, One solution