To choose the correct graph for the given system of equations:
[tex]\[
\begin{array}{l}
y + 2x = -1 \\
3y - x = 4
\end{array}
\][/tex]
we need to find the solution for the system and then verify which graph represents our solution.
Step-by-Step Solution:
1. Rewrite the first equation in slope-intercept form ( [tex]\( y = mx + b \)[/tex] ):
[tex]\[
y + 2x = -1
\][/tex]
Subtract [tex]\( 2x \)[/tex] from both sides:
[tex]\[
y = -2x - 1
\][/tex]
2. Rewrite the second equation in slope-intercept form:
[tex]\[
3y - x = 4
\][/tex]
Add [tex]\( x \)[/tex] to both sides:
[tex]\[
3y = x + 4
\][/tex]
Divide by 3:
[tex]\[
y = \frac{1}{3}x + \frac{4}{3}
\][/tex]
3. Identify the slopes and y-intercepts of both lines:
- For the first equation [tex]\( y = -2x - 1 \)[/tex]: the slope [tex]\( m = -2 \)[/tex] and the y-intercept is [tex]\( -1 \)[/tex].
- For the second equation [tex]\( y = \frac{1}{3}x + \frac{4}{3} \)[/tex]: the slope [tex]\( m = \frac{1}{3} \)[/tex] and the y-intercept is [tex]\( \frac{4}{3} \)[/tex].
4. Plot the lines:
- The first line [tex]\( y = -2x - 1 \)[/tex] has a negative slope and crosses the y-axis at [tex]\( -1 \)[/tex].
- The second line [tex]\( y = \frac{1}{3}x + \frac{4}{3} \)[/tex] has a positive slope and crosses the y-axis at [tex]\( \frac{4}{3} \)[/tex].
5. Find the point of intersection:
- The point where the two lines intersect is the solution to the system. Plugging in the solution deduced:
[tex]\[
(x, y) = (-1, 1)
\][/tex]
- This means the point (-1, 1) should be on both lines.
6. Verify the Intersection:
- Substitute [tex]\( x = -1 \)[/tex] into [tex]\( y = -2x - 1 \)[/tex]:
[tex]\[ y = -2(-1) - 1 = 2 - 1 = 1 \][/tex]
- Substitute [tex]\( x = -1 \)[/tex] into [tex]\( y = \frac{1}{3}x + \frac{4}{3} \)[/tex]:
[tex]\[ y = \frac{1}{3}(-1) + \frac{4}{3} = -\frac{1}{3} + \frac{4}{3} = \frac{3}{3} = 1 \][/tex]
Therefore, the solution to the system is [tex]\((x, y) = (-1, 1)\)[/tex].
To choose the correct graph, look for the graph where:
- One line has a slope of [tex]\(-2\)[/tex] and a y-intercept of [tex]\(-1\)[/tex].
- Another line has a slope of [tex]\(\frac{1}{3}\)[/tex] and a y-intercept of [tex]\(\frac{4}{3}\)[/tex].
- The point of intersection is [tex]\((-1, 1)\)[/tex].
Choose the graph matching these criteria.
