Question

write the equation of a line in standard form that models the graph below.
none of the answers
$y = \frac{5}{3}x - 3$
$5x - 3y = 9$
$y - 2 = \frac{5}{3}(x - 3)$

Explanation:

Step1: Identify two points on the line

From the graph, we can see that the line passes through \((0, -3)\) (the y - intercept) and \((3, 2)\) (we can find this by moving 3 units to the right and 5 units up from the y - intercept? Wait, let's calculate the slope first. The slope \(m=\frac{y_2 - y_1}{x_2 - x_1}\). Using \((0,-3)\) as \((x_1,y_1)\) and \((3,2)\) as \((x_2,y_2)\), \(m=\frac{2-(-3)}{3 - 0}=\frac{5}{3}\).

Step2: Write the slope - intercept form

The slope - intercept form of a line is \(y=mx + b\), where \(m\) is the slope and \(b\) is the y - intercept. We know \(m = \frac{5}{3}\) and \(b=-3\), so the equation in slope - intercept form is \(y=\frac{5}{3}x-3\). But we need to convert it to standard form \(Ax + By=C\), where \(A\), \(B\), and \(C\) are integers and \(A\geq0\).
Starting with \(y=\frac{5}{3}x-3\), multiply both sides by 3 to get rid of the fraction: \(3y = 5x-9\). Then, rearrange the terms to get \(5x-3y = 9\).
Let's check the options:

• Option \(y=\frac{5}{3}x - 3\) is in slope - intercept form, not standard form.
• Option \(5x-3y = 9\) is in standard form (\(A = 5\), \(B=-3\), \(C = 9\)) and when we derived the equation, we got \(5x-3y=9\).
• Option \(y - 2=\frac{5}{3}(x - 3)\) is in point - slope form.
• Option "None of the answers" is incorrect as we have found that \(5x-3y = 9\) is the correct standard form.

Answer:

\(5x - 3y=9\) (the option with \(5x - 3y = 9\))