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 line is $y = mx + b$, where $m$ is the slope and $b$ is the y - intercept. We know that the line passes through $(0,2)$, so when $x = 0$, $y=2$. Substituting into $y=mx + b$, we get $2=m(0)+b$, so $b = 2$.

Step2: Calculate the slope

We also know that the line passes through $(4,1)$ and $(0,2)$. The formula for slope $m=\frac{y_2 - y_1}{x_2 - x_1}$. Let $(x_1,y_1)=(0,2)$ and $(x_2,y_2)=(4,1)$. Then $m=\frac{1 - 2}{4 - 0}=\frac{-1}{4}=-\frac{1}{4}$.

Step3: 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 $x + 4y=8$)