Question

match each point of intersection with the system of equations whose solution is at that point.

\\(\

$$\begin{cases} y = -2x - 3 \\\\ y = 2x - 1 \\end{cases}$$

\\) \\(\
ightarrow\\)

\\(\

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

\\) \\(\
ightarrow\\)

\\(\

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

\\) \\(\
ightarrow\\)

\\(\

$$\begin{cases} y = 2x - 1 \\\\ y = -x + 2 \\end{cases}$$

\\) \\(\
ightarrow\\)

Explanation:

Identify coordinates of the intersection points

We first locate the coordinates of the labeled points of intersection on the coordinate plane:

• Point \(W\) is located at \((-2.5, 1.5)\).
• Point \(X\) is located at \((-1, 3)\).
• Point \(Y\) is located at \((1, 1)\).
• Point \(Z\) is located at \((-1, -2)\).

Match the first system of equations

Using the Systems of Linear Equations concept, we solve the first system:

$$
LATEXBLOCK0
$$

Setting the equations equal to each other:

$$ 2x - 1 = -2x - 3 \implies 4x = -2 \implies x = -0.5 $$

Substituting \(x = -0.5\) back into the second equation:

$$ y = 2(-0.5) - 1 = -2 $$

The solution is \((-0.5, -2)\). Looking at the graph, this corresponds to point \(Z\) at \((-1, -2)\) (re-verifying grid lines: \(Z\) is exactly at \(x = -1\), let's check the lines passing through \(Z\): the line \(y = -2x - 3\) at \(x = -1\) gives \(y = -1\), and \(y = 2x - 1\) at \(x = -1\) gives \(y = -3\). Let's re-examine the grid lines carefully.
The origin \((0,0)\) is marked.

• \(X\) is at \((-1, 3)\).
• \(Y\) is at \((1, 1)\).
• \(Z\) is at \((-1, -2)\).
• \(W\) is at \((-2.5, 1.5)\).

Let's test the points directly in the systems to find the correct matches.

Test point coordinates in the systems

Let's test each point in the given systems of equations:

1. For Point \(X(-1, 3)\):

• Test in \(y = x + 4\): \(3 = -1 + 4\) (True)
• Test in \(y = -x + 2\): \(3 = -(-1) + 2 = 3\) (True)
• Thus, \(X(-1, 3)\) is the solution to the system:

$$
LATEXBLOCK1
$$

1. For Point \(W(-2.5, 1.5)\):

• Test in \(y = -2x - 3\): \(1.5 = -2(-2.5) - 3 = 5 - 3 = 2

eq 1.5\).

• Let's re-read the coordinates of \(W\). It lies on the intersection of the red line \(y = x + 4\) and the purple line \(y = -2x - 3\).
• Let's solve:

$$ x + 4 = -2x - 3 \implies 3x = -7 \implies x = -\frac{7}{3} \approx -2.33 $$

$$ y = -\frac{7}{3} + 4 = \frac{5}{3} \approx 1.67 $$

This matches the visual position of \(W\). Thus, \(W\) is the solution to:

$$
LATEXBLOCK2
$$

1. For Point \(Y(1, 1)\):

• Test in \(y = 2x - 1\): \(1 = 2(1) - 1 = 1\) (True)
• Test in \(y = -x + 2\): \(1 = -1 + 2 = 1\) (True)
• Thus, \(Y(1, 1)\) is the solution to the system:

$$
LATEXBLOCK3
$$

1. For Point \(Z(-1, -2)\):

• Let's check the lines intersecting at \(Z\). The lines are the blue-grey line \(y = 2x - 1\) and the purple line \(y = -2x - 3\).
• Let's solve:

$$ 2x - 1 = -2x - 3 \implies 4x = -2 \implies x = -0.5 $$

$$ y = 2(-0.5) - 1 = -2 $$

The exact intersection is at \((-0.5, -2)\), which is labeled as \(Z\) on the graph. Thus, \(Z\) is the solution to:

$$
LATEXBLOCK4
$$

Answer:

• \(

$$\begin{cases} y = -2x - 3 \\ y = 2x - 1 \end{cases}$$

\longrightarrow \text{Z}\)

• \(

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

\longrightarrow \text{X}\)

• \(

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

\longrightarrow \text{W}\)

• \(

$$\begin{cases} y = 2x - 1 \\ y = -x + 2 \end{cases}$$

\longrightarrow \text{Y}\)