To determine which graph represents the system of equations, we need to analyze each equation and find their respective lines by transforming them into slope-intercept form ([tex]\( y = mx + b \)[/tex]). Here are the steps:
1. Equation 1: [tex]\(-2x + y = 10\)[/tex]
Solve for [tex]\( y \)[/tex]:
[tex]\[
y = 2x + 10
\][/tex]
This equation represents a line with a slope ([tex]\( m \)[/tex]) of 2 and a y-intercept ([tex]\( b \)[/tex]) of 10.
2. Equation 2: [tex]\( x + 2y = 5 \)[/tex]
Solve for [tex]\( y \)[/tex]:
[tex]\[
2y = -x + 5
\][/tex]
[tex]\[
y = -\frac{1}{2}x + \frac{5}{2}
\][/tex]
This equation represents a line with a slope ([tex]\( m \)[/tex]) of [tex]\(-\frac{1}{2}\)[/tex] and a y-intercept ([tex]\( b \)[/tex]) of [tex]\(\frac{5}{2}\)[/tex].
Next, we find the point of intersection of the two lines, which represents the solution to the system of equations. The solution for the system is the point where both equations are satisfied simultaneously. For our system, the solution is:
[tex]\[
(x, y) = (-3, 4)
\][/tex]
To visually verify this, we find that the lines' intersection at [tex]\((-3, 4)\)[/tex] must appear on the graph. We can now summarize:
- The line [tex]\( y = 2x + 10 \)[/tex] should pass through the points [tex]\((0, 10)\)[/tex] and [tex]\((2, 14)\)[/tex], illustrating a steep upward slope.
- The line [tex]\( y = -\frac{1}{2}x + \frac{5}{2} \)[/tex] should pass through the points [tex]\((0, \frac{5}{2})\)[/tex] and [tex]\((2, \frac{1}{2})\)[/tex], illustrating a downward slope.
- These lines should intersect at [tex]\((-3, 4)\)[/tex].
When identifying the correct graph, confirm the following:
- Both lines have the correct slopes and y-intercepts.
- They intersect precisely at the point [tex]\((-3, 4)\)[/tex].
Thus, the graph which correctly represents these characteristics is the one where:
- The first line has a positive slope and intersects the y-axis at 10.
- The second line has a negative slope ([tex]\(-\frac{1}{2}\)[/tex]) and intersects the y-axis at [tex]\(\frac{5}{2}\)[/tex].
- The lines intersect at the point [tex]\((-3, 4)\)[/tex].
