Question

1. draw lines to match each equation on the left to its number of solutions on the right.

-2(3x - 1) = -6x - 1
2(3x - 1) = 6x - 2
2(3x - 1) = -6x - 2
one solution
no solution
infinitely many solutions

1. solve the equation \\(\frac{2}{3}x - \frac{1}{5}x = x - 1\\).

Explanation:

For Question 6:

Step1: Simplify first equation

Expand left side: $-2(3x-1) = -6x + 2$
Set equal to right side: $-6x + 2 = -6x - 1$
Cancel $-6x$: $2 = -1$ (false)

Step2: Simplify second equation

Expand left side: $2(3x-1) = 6x - 2$
Set equal to right side: $6x - 2 = 6x - 2$ (always true)

Step3: Simplify third equation

Expand left side: $2(3x-1) = 6x - 2$
Set equal to right side: $6x - 2 = -6x - 2$
Solve for $x$: $12x = 0 \implies x=0$ (single solution)

Step1: Combine like terms (left side)

Find common denominator 15: $\frac{2}{3}x - \frac{1}{5}x = \frac{10}{15}x - \frac{3}{15}x = \frac{7}{15}x$

Step2: Isolate $x$ terms

Subtract $x$ from both sides: $\frac{7}{15}x - x = -1$
Convert $x$ to $\frac{15}{15}x$: $\frac{7}{15}x - \frac{15}{15}x = -\frac{8}{15}x = -1$

Step3: Solve for $x$

Multiply both sides by $-\frac{15}{8}$: $x = (-1) \times (-\frac{15}{8}) = \frac{15}{8}$

Answer:

$-2(3x - 1) = -6x - 1$ → no solution
$2(3x - 1) = 6x - 2$ → infinitely many solutions
$2(3x - 1) = -6x - 2$ → one solution

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For Question 7: