Question

which set of statements shows that the point $(-3.5, -4)$ is the point where the two lines intersect?

a. $\

$$\begin{cases}-2(-3.5) + 5(-4) = -13\\\\12(-4) = 4(-3.5) - 34\\end{cases}$$

$

b. $\

$$\begin{cases}-2(-3.5) + 5(-3.5) = -13\\\\12(-4) = 4(-4) - 34\\end{cases}$$

$

c. $\

$$\begin{cases}-2(-4) + 5(-3.5) = -13\\\\12(-4) = 4(-3.5) - 34\\end{cases}$$

$

d. $\

$$\begin{cases}-2(-3.5) + 5(-4) = -13\\\\12(-3.5) = 4(-4) - 34\\end{cases}$$

$

Explanation:

To determine if the point \((-3.5, -4)\) is the intersection of two lines, we need to check if it satisfies both equations of the lines. For a point \((x, y)\) to be the intersection, it must satisfy both linear equations. Here, \(x = -3.5\) and \(y = -4\). We will substitute these values into each pair of equations in the options.

Step 1: Analyze Option A

First equation: \(-2x + 5y = -13\) (assuming the form, since we substitute \(x = -3.5\) and \(y = -4\))
Substitute \(x = -3.5\) and \(y = -4\):
\[
-2(-3.5) + 5(-4) = 7 - 20 = -13
\]
Second equation: \(12y = 4x - 34\) (assuming the form)
Substitute \(x = -3.5\) and \(y = -4\):
\[
12(-4) = -48 \quad \text{and} \quad 4(-3.5) - 34 = -14 - 34 = -48
\]
So, \(-48 = -48\). Both equations are satisfied.

Step 2: Analyze Option B

First equation: \(-2(-3.5) + 5(-3.5) = 7 - 17.5 = -10.5
eq -13\). So, this option is incorrect.

Step 3: Analyze Option C

First equation: \(-2(-4) + 5(-3.5) = 8 - 17.5 = -9.5
eq -13\). So, this option is incorrect.

Step 4: Analyze Option D

Second equation: \(12(-3.5) = -42\) and \(4(-4) - 34 = -16 - 34 = -50\). Since \(-42
eq -50\), this option is incorrect.

Answer:

A. \(

$$\begin{cases}-2(-3.5) + 5(-4) = -13 \\ 12(-4) = 4(-3.5) - 34\end{cases}$$

\)