To solve the system of equations [tex]\(\begin{array}{l}
y = 2x - 3 \\
y = -2x + 5
\end{array}\)[/tex] by graphing, follow these steps:
1. Graph the first equation [tex]\(y = 2x - 3\)[/tex]:
- Identify the y-intercept, which is the point where [tex]\(x = 0\)[/tex]. For the equation [tex]\(y = 2x - 3\)[/tex], the y-intercept is [tex]\((0, -3)\)[/tex].
- Identify another point by choosing a convenient value for [tex]\(x\)[/tex]. For example, if [tex]\(x = 1\)[/tex],
[tex]\[
y = 2(1) - 3 = 2 - 3 = -1
\][/tex]
So, another point is [tex]\((1, -1)\)[/tex].
- Plot the points [tex]\((0, -3)\)[/tex] and [tex]\((1, -1)\)[/tex] on the graph and draw a line through them.
2. Graph the second equation [tex]\(y = -2x + 5\)[/tex]:
- Identify the y-intercept, which is the point where [tex]\(x = 0\)[/tex]. For the equation [tex]\(y = -2x + 5\)[/tex], the y-intercept is [tex]\((0, 5)\)[/tex].
- Identify another point by choosing a convenient value for [tex]\(x\)[/tex]. For example, if [tex]\(x = 1\)[/tex],
[tex]\[
y = -2(1) + 5 = -2 + 5 = 3
\][/tex]
So, another point is [tex]\((1, 3)\)[/tex].
- Plot the points [tex]\((0, 5)\)[/tex] and [tex]\((1, 3)\)[/tex] on the graph and draw a line through them.
3. Find the point of intersection:
- The point where the two lines intersect is the solution to the system of equations.
Upon graphing both lines, you'll see that they intersect at the point [tex]\((2, 1)\)[/tex].
Thus, the solution to the system of equations is:
[tex]\[ \boxed{(2, 1)} \][/tex]
The solution matches option D: [tex]\((2, 1)\)[/tex].
