Question

here is a graph that represents one equation in a system of equations. write a second equation whose graph goes through (0, 2) so that the system has one solution: (4, 1). the equation on the graph is 4y = -3x + 16.

Explanation:

Step1: Recall slope-intercept form

The slope - intercept form of a linear equation is \(y = mx + b\), where \(m\) is the slope and \(b\) is the \(y\) - intercept. We know that the line passes through the points \((0,2)\) and \((4,1)\).

Step2: Calculate the slope (\(m\))

The formula for the slope \(m\) between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(m=\frac{y_2 - y_1}{x_2 - x_1}\). Substituting \(x_1 = 0,y_1 = 2,x_2=4,y_2 = 1\) into the formula, we get \(m=\frac{1 - 2}{4 - 0}=\frac{-1}{4}=-\frac{1}{4}\).

Step3: Determine the \(y\) - intercept (\(b\))

Since the line passes through \((0,2)\), when \(x = 0\), \(y=b\). So \(b = 2\).

Step4: Write the equation

Using the slope - intercept form \(y=mx + b\) with \(m=-\frac{1}{4}\) and \(b = 2\), the equation is \(y=-\frac{1}{4}x+2\). We can also multiply through by 4 to get \(4y=-x + 8\) or \(x+4y=8\).

Answer:

\(y =-\frac{1}{4}x + 2\) (or equivalent forms like \(4y=-x + 8\) or \(x + 4y=8\))