To estimate the solution to the system of the linear equations \( y = 5x + 10 \) and \( y = -2x + 3 \), we need to find the point where these two lines intersect.
-
Set the equations equal to each other since they both equal \( y \): \[ 5x + 10 = -2x + 3 \]
-
Solve for \( x \):
- Add \( 2x \) to both sides: \[ 5x + 2x + 10 = 3 \] \[ 7x + 10 = 3 \]
- Subtract 10 from both sides: \[ 7x = 3 - 10 \] \[ 7x = -7 \]
- Divide by 7: \[ x = -1 \]
-
Substitute \( x = -1 \) into one of the original equations to find \( y \). We'll use \( y = 5x + 10 \): \[ y = 5(-1) + 10 \] \[ y = -5 + 10 \] \[ y = 5 \]
The solution to the system of equations, which is the point of intersection, is: \[ \boxed{(-1, 5)} \]
If you were to graph these equations, you would see that their lines intersect at the point \((-1, 5)\).