Question
Which graph represents this system?
2 x minus 5 y = negative 5. y = two-fifths x + 1.
On a coordinate plane, 2 lines go through points (negative 5, negative 1) and (0, 1).
On a coordinate plane, a line goes through (0, 1) and (5, 3) and another goes through (0, negative 5) and (5, negative 3).
On a coordinate plane, a line goes through (0, 1) and (5, 3) and another goes through (0, 1) and (5, negative 1).
On a coordinate plane, a line goes through (0, 1) and (negative 5, negative 1) and another goes through (negative 5, negative 3) and (0, negative 5).
2 x minus 5 y = negative 5. y = two-fifths x + 1.
On a coordinate plane, 2 lines go through points (negative 5, negative 1) and (0, 1).
On a coordinate plane, a line goes through (0, 1) and (5, 3) and another goes through (0, negative 5) and (5, negative 3).
On a coordinate plane, a line goes through (0, 1) and (5, 3) and another goes through (0, 1) and (5, negative 1).
On a coordinate plane, a line goes through (0, 1) and (negative 5, negative 1) and another goes through (negative 5, negative 3) and (0, negative 5).
Answers
GPT-4o mini
To analyze which graph represents the given system of equations, we can rewrite and identify the lines from each equation.
The first equation is:
\[ 2x - 5y = -5 \]
To find the slope-intercept form (y = mx + b), we can isolate \( y \):
\[ -5y = -2x - 5 \]
\[ y = \frac{2}{5}x + 1 \]
This tells us that the line has a slope of \( \frac{2}{5} \) and a y-intercept of \( 1 \).
The second equation is already in slope-intercept form:
\[ y = \frac{2}{5}x + 1 \]
This line is identical to the first line.
Since both lines represent the same equation, we should look for the pair of lines passing through (0, 1) and having the same slope.
Now let's review the options presented:
1. **On a coordinate plane, two lines go through points (negative 5, negative 1) and (0, 1).**
- This line has a positive slope as it is rising from left to right.
2. **On a coordinate plane, a line goes through (0, 1) and (5, 3) and another goes through (0, negative 5) and (5, negative 3).**
- The first line's slope is \( \frac{3 - 1}{5 - 0} = \frac{2}{5} \) and the second line has a negative slope.
3. **On a coordinate plane, a line goes through (0, 1) and (5, 3) and another goes through (0, 1) and (5, negative 1).**
- Here, the first line has a slope of \( \frac{3 - 1}{5 - 0} = \frac{2}{5} \) and the second line also starts at (0, 1).
4. **On a coordinate plane, a line goes through (0, 1) and (negative 5, negative 1) and another goes through (negative 5, negative 3) and (0, negative 5).**
- The first line has a slope that is positive as it passes through points in the first and fourth quadrants.
Thus, the option that satisfies the requirement of having two identical lines is:
**On a coordinate plane, a line goes through (0, 1) and (5, 3) and another goes through (0, 1) and (5, negative 1).**
This is the correct option as both lines share the point (0, 1) and represent either the same line or parallel lines that intersect at (0, 1).
The first equation is:
\[ 2x - 5y = -5 \]
To find the slope-intercept form (y = mx + b), we can isolate \( y \):
\[ -5y = -2x - 5 \]
\[ y = \frac{2}{5}x + 1 \]
This tells us that the line has a slope of \( \frac{2}{5} \) and a y-intercept of \( 1 \).
The second equation is already in slope-intercept form:
\[ y = \frac{2}{5}x + 1 \]
This line is identical to the first line.
Since both lines represent the same equation, we should look for the pair of lines passing through (0, 1) and having the same slope.
Now let's review the options presented:
1. **On a coordinate plane, two lines go through points (negative 5, negative 1) and (0, 1).**
- This line has a positive slope as it is rising from left to right.
2. **On a coordinate plane, a line goes through (0, 1) and (5, 3) and another goes through (0, negative 5) and (5, negative 3).**
- The first line's slope is \( \frac{3 - 1}{5 - 0} = \frac{2}{5} \) and the second line has a negative slope.
3. **On a coordinate plane, a line goes through (0, 1) and (5, 3) and another goes through (0, 1) and (5, negative 1).**
- Here, the first line has a slope of \( \frac{3 - 1}{5 - 0} = \frac{2}{5} \) and the second line also starts at (0, 1).
4. **On a coordinate plane, a line goes through (0, 1) and (negative 5, negative 1) and another goes through (negative 5, negative 3) and (0, negative 5).**
- The first line has a slope that is positive as it passes through points in the first and fourth quadrants.
Thus, the option that satisfies the requirement of having two identical lines is:
**On a coordinate plane, a line goes through (0, 1) and (5, 3) and another goes through (0, 1) and (5, negative 1).**
This is the correct option as both lines share the point (0, 1) and represent either the same line or parallel lines that intersect at (0, 1).