Question
Use the image to answer the question.
A coordinate plane with four quadrants shows the x- and y-axes ranging from negative 5 to 5 in increments of 1. A solid line and a dotted line intersect each other. The equation of the solid line is x minus 5 y equals 3. The equation of the dotted line is 3 x minus 2 y equals negative 4. The intersection of both lines is shown at negative 2 on the x-axis and negative 1 on the y-axis in quadrant 3.
Review the graphs of a system of two linear equations in two variables: x−5y=7
and 3x−2y=−4
. Find the solution to both equations.
(1 point)
The intersection point is (
).
A coordinate plane with four quadrants shows the x- and y-axes ranging from negative 5 to 5 in increments of 1. A solid line and a dotted line intersect each other. The equation of the solid line is x minus 5 y equals 3. The equation of the dotted line is 3 x minus 2 y equals negative 4. The intersection of both lines is shown at negative 2 on the x-axis and negative 1 on the y-axis in quadrant 3.
Review the graphs of a system of two linear equations in two variables: x−5y=7
and 3x−2y=−4
. Find the solution to both equations.
(1 point)
The intersection point is (
).
Answers
Answer
can you make it easier
Answered by
GPT-4o mini
To find the solution to the system of equations represented by the lines \( x - 5y = 7 \) and \( 3x - 2y = -4 \), we need to solve the system.
Starting with the first equation:
1. \( x - 5y = 7 \)
We can express \( x \) in terms of \( y \):
\[
x = 5y + 7
\]
Now substituting \( x \) into the second equation:
2. \( 3x - 2y = -4 \)
Substituting \( x = 5y + 7 \):
\[
3(5y + 7) - 2y = -4
\]
Expanding this:
\[
15y + 21 - 2y = -4
\]
Combining like terms:
\[
13y + 21 = -4
\]
Now, isolating \( y \):
\[
13y = -4 - 21
\]
\[
13y = -25
\]
\[
y = -\frac{25}{13}
\]
Now that we have \( y \), we substitute back to find \( x \):
\[
x = 5\left(-\frac{25}{13}\right) + 7
\]
\[
x = -\frac{125}{13} + 7
\]
\[
x = -\frac{125}{13} + \frac{91}{13}
\]
\[
x = -\frac{34}{13}
\]
Thus, the solution to the system of equations is:
\[
\left(-\frac{34}{13}, -\frac{25}{13}\right)
\]
As a decimal approximation, this would be approximately:
\[
(-2.615, -1.923)
\]
Therefore, the intersection point is:
\[
\left(-\frac{34}{13}, -\frac{25}{13}\right)
\]
Starting with the first equation:
1. \( x - 5y = 7 \)
We can express \( x \) in terms of \( y \):
\[
x = 5y + 7
\]
Now substituting \( x \) into the second equation:
2. \( 3x - 2y = -4 \)
Substituting \( x = 5y + 7 \):
\[
3(5y + 7) - 2y = -4
\]
Expanding this:
\[
15y + 21 - 2y = -4
\]
Combining like terms:
\[
13y + 21 = -4
\]
Now, isolating \( y \):
\[
13y = -4 - 21
\]
\[
13y = -25
\]
\[
y = -\frac{25}{13}
\]
Now that we have \( y \), we substitute back to find \( x \):
\[
x = 5\left(-\frac{25}{13}\right) + 7
\]
\[
x = -\frac{125}{13} + 7
\]
\[
x = -\frac{125}{13} + \frac{91}{13}
\]
\[
x = -\frac{34}{13}
\]
Thus, the solution to the system of equations is:
\[
\left(-\frac{34}{13}, -\frac{25}{13}\right)
\]
As a decimal approximation, this would be approximately:
\[
(-2.615, -1.923)
\]
Therefore, the intersection point is:
\[
\left(-\frac{34}{13}, -\frac{25}{13}\right)
\]
Answered by
GPT-4o mini
Sure! Let's simplify the process of finding the intersection point of the two equations, \(x - 5y = 7\) and \(3x - 2y = -4\).
### Step 1: Solve the first equation for \(x\)
From the first equation:
\[
x - 5y = 7
\]
We can express \(x\) in terms of \(y\):
\[
x = 5y + 7
\]
### Step 2: Substitute into the second equation
Next, we take the second equation, \(3x - 2y = -4\), and substitute \(x\) from the first equation:
\[
3(5y + 7) - 2y = -4
\]
### Step 3: Simplify
Now we will simplify the equation:
\[
15y + 21 - 2y = -4
\]
\[
13y + 21 = -4
\]
### Step 4: Isolate \(y\)
Now isolate \(y\):
\[
13y = -4 - 21
\]
\[
13y = -25
\]
\[
y = -\frac{25}{13}
\]
### Step 5: Substitute to find \(x\)
Now substitute \(y\) back into the equation for \(x\):
\[
x = 5\left(-\frac{25}{13}\right) + 7
\]
\[
x = -\frac{125}{13} + \frac{91}{13}
\]
\[
x = -\frac{34}{13}
\]
### Final Answer
So the intersection point of the two lines is:
\[
\left(-\frac{34}{13}, -\frac{25}{13}\right)
\]
As approximate decimal values, this is about:
\[
(-2.615, -1.923)
\]
### Conclusion
Thus, the intersection point is:
\[
\left(-\frac{34}{13}, -\frac{25}{13}\right)
\]
If you just need the coordinates, it can be written as:
\[
(-2.615, -1.923)
\]
Let me know if you need anything else or further clarification!
### Step 1: Solve the first equation for \(x\)
From the first equation:
\[
x - 5y = 7
\]
We can express \(x\) in terms of \(y\):
\[
x = 5y + 7
\]
### Step 2: Substitute into the second equation
Next, we take the second equation, \(3x - 2y = -4\), and substitute \(x\) from the first equation:
\[
3(5y + 7) - 2y = -4
\]
### Step 3: Simplify
Now we will simplify the equation:
\[
15y + 21 - 2y = -4
\]
\[
13y + 21 = -4
\]
### Step 4: Isolate \(y\)
Now isolate \(y\):
\[
13y = -4 - 21
\]
\[
13y = -25
\]
\[
y = -\frac{25}{13}
\]
### Step 5: Substitute to find \(x\)
Now substitute \(y\) back into the equation for \(x\):
\[
x = 5\left(-\frac{25}{13}\right) + 7
\]
\[
x = -\frac{125}{13} + \frac{91}{13}
\]
\[
x = -\frac{34}{13}
\]
### Final Answer
So the intersection point of the two lines is:
\[
\left(-\frac{34}{13}, -\frac{25}{13}\right)
\]
As approximate decimal values, this is about:
\[
(-2.615, -1.923)
\]
### Conclusion
Thus, the intersection point is:
\[
\left(-\frac{34}{13}, -\frac{25}{13}\right)
\]
If you just need the coordinates, it can be written as:
\[
(-2.615, -1.923)
\]
Let me know if you need anything else or further clarification!