Question
or Solve this system of equations by graphing. First graph the equations, and then type the solution. y= 4/5 x+1 y= – 3/5 x–6, write a graph
Answers
GPT-4o mini
To solve the system of equations \(y = \frac{4}{5} x + 1\) and \(y = -\frac{3}{5} x - 6\) by graphing, we follow these steps:
1. **Graph the first equation**: \(y = \frac{4}{5} x + 1\)
- This equation has a slope of \(\frac{4}{5}\) and a y-intercept of \(1\).
- To plot it, start at the y-intercept (0, 1). From there, use the slope to find another point: rise 4 and run 5 (up 4 and right 5).
- This gives us the point \((5, 5)\).
- We can draw the line through these points.
2. **Graph the second equation**: \(y = -\frac{3}{5} x - 6\)
- This equation has a slope of \(-\frac{3}{5}\) and a y-intercept of \(-6\).
- Start at the y-intercept (0, -6). From there, use the slope: down 3 and right 5 (the negative slope means you move down).
- This gives us the point \((5, -9)\).
- Draw the line through these points.
3. **Find the intersection point**:
- After graphing both lines, look for the point where they intersect. This point is the solution to the system of equations.
### Solution
The lines will intersect at a specific point, which can also be calculated algebraically by setting the two equations equal to each other:
\[
\frac{4}{5}x + 1 = -\frac{3}{5}x - 6
\]
To solve for \(x\):
1. Combine like terms:
\[
\frac{4}{5}x + \frac{3}{5}x = -6 - 1
\]
\[
\frac{7}{5}x = -7
\]
2. Multiply both sides by \(\frac{5}{7}\):
\[
x = -5
\]
Now substitute \(x = -5\) back into either original equation to find \(y\). Let’s use the first one:
\[
y = \frac{4}{5}(-5) + 1 = -4 + 1 = -3
\]
### Final Answer
The solution to the system of equations is:
\[
\boxed{(-5, -3)}
\]
You can graph this on a graphing tool to visualize the solution. If you have graphing software or graph paper, plot both lines to see where they intersect at the point \((-5, -3)\).
1. **Graph the first equation**: \(y = \frac{4}{5} x + 1\)
- This equation has a slope of \(\frac{4}{5}\) and a y-intercept of \(1\).
- To plot it, start at the y-intercept (0, 1). From there, use the slope to find another point: rise 4 and run 5 (up 4 and right 5).
- This gives us the point \((5, 5)\).
- We can draw the line through these points.
2. **Graph the second equation**: \(y = -\frac{3}{5} x - 6\)
- This equation has a slope of \(-\frac{3}{5}\) and a y-intercept of \(-6\).
- Start at the y-intercept (0, -6). From there, use the slope: down 3 and right 5 (the negative slope means you move down).
- This gives us the point \((5, -9)\).
- Draw the line through these points.
3. **Find the intersection point**:
- After graphing both lines, look for the point where they intersect. This point is the solution to the system of equations.
### Solution
The lines will intersect at a specific point, which can also be calculated algebraically by setting the two equations equal to each other:
\[
\frac{4}{5}x + 1 = -\frac{3}{5}x - 6
\]
To solve for \(x\):
1. Combine like terms:
\[
\frac{4}{5}x + \frac{3}{5}x = -6 - 1
\]
\[
\frac{7}{5}x = -7
\]
2. Multiply both sides by \(\frac{5}{7}\):
\[
x = -5
\]
Now substitute \(x = -5\) back into either original equation to find \(y\). Let’s use the first one:
\[
y = \frac{4}{5}(-5) + 1 = -4 + 1 = -3
\]
### Final Answer
The solution to the system of equations is:
\[
\boxed{(-5, -3)}
\]
You can graph this on a graphing tool to visualize the solution. If you have graphing software or graph paper, plot both lines to see where they intersect at the point \((-5, -3)\).