Question

the tables represent two linear functions in a system. what is the solution to this system? options: (-11/3, -25), (-14/3, -54), (-13, -50), (-14, -54)

Explanation:

Step1: Find the slope - intercept form of the line.

The slope $m$ of a line passing through two points $(x_1,y_1)$ and $(x_2,y_2)$ is given by $m=\frac{y_2 - y_1}{x_2 - x_1}$. Let's take two points from the first table, say $(x_1 = 3,y_1=14)$ and $(x_2 = 0,y_2 = 2)$. Then $m=\frac{2 - 14}{0 - 3}=\frac{- 12}{-3}=4$. Using the point - slope form $y - y_1=m(x - x_1)$ with the point $(0,2)$ (so $x_1 = 0,y_1 = 2$ and $m = 4$), we get $y=4x + 2$.

Step2: Check the second table for the same relationship.

Let's take two points from the second table, say $(x_1 = 3,y_1=-3)$ and $(x_2 = 0,y_2=-12)$. The slope $m=\frac{-12+3}{0 - 3}=\frac{-9}{-3}=3$. Using the point - slope form with the point $(0,-12)$ (so $x_1 = 0,y_1=-12$ and $m = 3$), we get $y = 3x-12$.

Step3: Solve the system of equations.

We have the system of equations

$$\begin{cases}y=4x + 2\\y=3x-12\end{cases}$$

. Since both expressions equal $y$, we can set them equal to each other: $4x+2=3x - 12$.

Step4: Isolate the variable $x$.

Subtract $3x$ from both sides: $4x-3x+2=3x-3x - 12$, which simplifies to $x+2=-12$. Then subtract 2 from both sides: $x=-12 - 2=-14$.

Step5: Find the value of $y$.

Substitute $x=-14$ into $y = 4x+2$. So $y=4\times(-14)+2=-56 + 2=-54$.

Answer:

$(-14,-54)$