Question
Which graph shows the solution to the system of linear equations?
y = 4x
y = x − 1
a coordinate grid with one line that passes through the points 0 comma 0 and 1 comma 2 and another line that passes through the points 0 comma negative 2 and 1 comma negative 1
a coordinate grid with one line that passes through the points 0 comma 2 and 1 comma 3 and another line that passes through the points 0 comma 0 and 1 comma 2
a coordinate grid with one line that passes through the points 0 comma 0 and 1 comma 4 and another line that passes through the points 0 comma 3 and negative 1 comma negative 1
a coordinate grid with one line that passes through the points 0 comma 0 and 1 comma 4 and another line that passes through the points 0 comma negative 1 and 2 comma 1
y = 4x
y = x − 1
a coordinate grid with one line that passes through the points 0 comma 0 and 1 comma 2 and another line that passes through the points 0 comma negative 2 and 1 comma negative 1
a coordinate grid with one line that passes through the points 0 comma 2 and 1 comma 3 and another line that passes through the points 0 comma 0 and 1 comma 2
a coordinate grid with one line that passes through the points 0 comma 0 and 1 comma 4 and another line that passes through the points 0 comma 3 and negative 1 comma negative 1
a coordinate grid with one line that passes through the points 0 comma 0 and 1 comma 4 and another line that passes through the points 0 comma negative 1 and 2 comma 1
Answers
GPT-4o mini
To determine which graph shows the solution to the system of equations \(y = 4x\) and \(y = x - 1\), we can analyze both equations:
1. **Equation 1: \(y = 4x\)**
- This is a line with a slope of 4 and a y-intercept of 0. It passes through the origin (0,0) and increases steeply.
2. **Equation 2: \(y = x - 1\)**
- This is a line with a slope of 1 and a y-intercept of -1. The line passes through the point (0, -1) and increases less steeply than the first line.
Next, we can find the intersection (where they will intersect) by setting the two equations equal to each other:
\[
4x = x - 1
\]
Subtract \(x\) from both sides:
\[
4x - x = -1
\]
\[
3x = -1
\]
\[
x = -\frac{1}{3}
\]
Now substitute \(x = -\frac{1}{3}\) back into one of the original equations (we can use \(y = 4x\)):
\[
y = 4(-\frac{1}{3}) = -\frac{4}{3}
\]
Thus, the intersection point is \(\left(-\frac{1}{3}, -\frac{4}{3}\right)\).
Now we can evaluate the options given based on the behavior of the lines:
1. **First option:** One line passes through (0,0) and (1,2); this corresponds to \(y = 2x\), which does not match.
2. **Second option:** One line passes through (0,2) and (1,3); the other through (0,0) and (1,2); these do not match either.
3. **Third option:** One line passes through (0,0) and (1,4); the other through (0,3) and (-1,-1); these do not match.
4. **Fourth option:** One line passes through (0,0) and (1,4) (this corresponds to \(y=4x\)), while the other passes through (0,-1) and (2,1) (which corresponds to \(y=x-1\)).
Given that the fourth option has one line corresponding to \(y=4x\) and the second line \(y=x-1\), this option is consistent with our original equations.
Therefore, the fourth graph is the one that shows the solution to the system of linear equations.
1. **Equation 1: \(y = 4x\)**
- This is a line with a slope of 4 and a y-intercept of 0. It passes through the origin (0,0) and increases steeply.
2. **Equation 2: \(y = x - 1\)**
- This is a line with a slope of 1 and a y-intercept of -1. The line passes through the point (0, -1) and increases less steeply than the first line.
Next, we can find the intersection (where they will intersect) by setting the two equations equal to each other:
\[
4x = x - 1
\]
Subtract \(x\) from both sides:
\[
4x - x = -1
\]
\[
3x = -1
\]
\[
x = -\frac{1}{3}
\]
Now substitute \(x = -\frac{1}{3}\) back into one of the original equations (we can use \(y = 4x\)):
\[
y = 4(-\frac{1}{3}) = -\frac{4}{3}
\]
Thus, the intersection point is \(\left(-\frac{1}{3}, -\frac{4}{3}\right)\).
Now we can evaluate the options given based on the behavior of the lines:
1. **First option:** One line passes through (0,0) and (1,2); this corresponds to \(y = 2x\), which does not match.
2. **Second option:** One line passes through (0,2) and (1,3); the other through (0,0) and (1,2); these do not match either.
3. **Third option:** One line passes through (0,0) and (1,4); the other through (0,3) and (-1,-1); these do not match.
4. **Fourth option:** One line passes through (0,0) and (1,4) (this corresponds to \(y=4x\)), while the other passes through (0,-1) and (2,1) (which corresponds to \(y=x-1\)).
Given that the fourth option has one line corresponding to \(y=4x\) and the second line \(y=x-1\), this option is consistent with our original equations.
Therefore, the fourth graph is the one that shows the solution to the system of linear equations.