Question

a line that passes through the points (-4, 10) and (-1, 5) can be represented by the equation $y = -\frac{5}{3}(x - 2)$. which equations also represent this line? select three options.
$\square\\ y = -\frac{5}{3}x - 2$
$\square\\ y = -\frac{5}{3}x + \frac{10}{3}$
$\square\\ 3y = -5x + 10$
$\square\\ 3x + 15y = 30$
$\square\\ 5x + 3y = 10$

Explanation:

Step1: Simplify the given equation

The given equation is \( y = -\frac{5}{3}(x - 2) \). Distribute the \( -\frac{5}{3} \):
\( y = -\frac{5}{3}x+\frac{10}{3} \)

Step2: Check each option

• Option 1: \( y = -\frac{5}{3}x - 2 \). The y - intercept here is \( - 2 \), but from our simplified equation, the y - intercept is \( \frac{10}{3}\approx3.33 \), so this is incorrect.
• Option 2: \( y = -\frac{5}{3}x+\frac{10}{3} \). This matches our simplified equation from Step 1, so this is correct.
• Option 3: Start with \( y = -\frac{5}{3}x+\frac{10}{3} \). Multiply both sides by 3: \( 3y=- 5x + 10 \). This is the same as the option, so this is correct.
• Option 4: Start with \( 3y=-5x + 10 \), we can rewrite it as \( 5x+3y = 10 \), not \( 3x + 15y=30 \). Let's check by plugging in a point, say \( x=-4,y = 10 \). For \( 3x+15y \), we have \( 3\times(-4)+15\times10=-12 + 150 = 138

eq30 \), so this is incorrect.

• Option 5: From \( y = -\frac{5}{3}x+\frac{10}{3} \), multiply both sides by 3: \( 3y=-5x + 10 \), which can be rewritten as \( 5x+3y = 10 \). This matches the option, so this is correct.

Answer:

B. \( y = -\frac{5}{3}x+\frac{10}{3} \), C. \( 3y=-5x + 10 \), E. \( 5x + 3y=10 \)