To analyze the system of equations given:
- \( y = 4x - 10 \)
- \( y = -2x + 8 \)
Step 1: Graph the equations
Equation 1: \( y = 4x - 10 \)
- This is in slope-intercept form \( y = mx + b \) where \( m = 4 \) (slope) and \( b = -10 \) (y-intercept).
- The graph will cross the y-axis at \((0, -10)\).
- The slope \(4\) means for every 1 unit increase in \(x\), \(y\) increases by \(4\).
Equation 2: \( y = -2x + 8 \)
- Similarly, this is in slope-intercept form where \( m = -2 \) and \( b = 8 \).
- It will cross the y-axis at \((0, 8)\).
- The slope \(-2\) means for every 1 unit increase in \(x\), \(y\) decreases by \(2\).
Step 2: Find the solution
To find the solution to the system, set the two equations equal to each other:
\[ 4x - 10 = -2x + 8 \]
Now solve for \(x\):
-
Add \(2x\) to both sides: \[ 4x + 2x - 10 = 8 \] \[ 6x - 10 = 8 \]
-
Add \(10\) to both sides: \[ 6x = 18 \]
-
Divide by \(6\): \[ x = 3 \]
Now, substitute \(x = 3\) back into one of the original equations to find \(y\). Using \(y = 4x - 10\):
\[ y = 4(3) - 10 = 12 - 10 = 2 \]
Final Solution
The solution to the system of equations is:
\[ (x, y) = (3, 2) \]
Verification
To verify, substitute \(x = 3\) into the second equation \(y = -2x + 8\):
\[ y = -2(3) + 8 = -6 + 8 = 2 \]
Both equations yield \(y = 2\) at \(x = 3\).
Thus, the point of intersection is \((3, 2)\), which is the solution to the system of equations.