To find the solution of the system of equations given by:
- \( y = 4x - 10 \)
- \( y = -2x + 8 \)
we can find the point at which these two lines intersect. This point is the solution to the system of equations.
Step 1: Set the equations equal to each other
Since both equations are equal to \( y \), we can set them equal to each other:
\[ 4x - 10 = -2x + 8 \]
Step 2: Solve for \( x \)
- Rearrange the equation to isolate \( x \). First, add \( 2x \) to both sides:
\[ 4x + 2x - 10 = 8 \] \[ 6x - 10 = 8 \]
- Now, add 10 to both sides:
\[ 6x = 18 \]
- Finally, divide by 6:
\[ x = 3 \]
Step 3: Substitute \( x \) back to find \( y \)
Now that we have \( x = 3 \), we can substitute this value back into either equation to find \( y \). Let's use the first equation:
\[ y = 4(3) - 10 \] \[ y = 12 - 10 \] \[ y = 2 \]
Conclusion
The solution to the system of equations is:
\[ (x, y) = (3, 2) \]
Verification
To verify, substitute \( x = 3 \) into the second equation as well:
\[ y = -2(3) + 8 \] \[ y = -6 + 8 \] \[ y = 2 \]
Both equations give the same result for \( y \), confirming that the solution \( (3, 2) \) is correct. The point where the two lines intersect—hence the solution to the system of equations— is \( (3, 2) \).