To solve the system of equations given by \( y = -x + 6 \) and \( y = 3x - 2 \), we can estimate the solution graphically or find the point of intersection algebraically. Here, I'll describe how to do both.
Step 1: Graph the equations
-
Equation 1: \( y = -x + 6 \)
- This is a line with a slope of \(-1\) and a y-intercept of \(6\).
- The y-intercept means the line crosses the y-axis at the point (0, 6).
- Another point on the line can be found by choosing \(x = 0\): \[ y = -0 + 6 \Rightarrow y = 6 \text{ (point: (0, 6))} \]
- Choosing \(x = 6\) gives \(y = -6 + 6 = 0 \text{ (point: (6, 0))}\).
-
Equation 2: \( y = 3x - 2 \)
- This is a line with a slope of \(3\) and a y-intercept of \(-2\).
- It crosses the y-axis at the point (0, -2).
- Choosing \(x = 0\) gives: \[ y = 3(0) - 2 \Rightarrow y = -2 \text{ (point: (0, -2))} \]
- Choosing \(x = 1\) gives: \[ y = 3(1) - 2 = 1 \text{ (point: (1, 1))} \]
Step 2: Find the intersection point
You can find the intersection point algebraically by setting the two equations equal to each other:
\[ -x + 6 = 3x - 2 \]
Step 3: Solve for \(x\)
- Rearrange the equation: \[ 6 + 2 = 3x + x \] \[ 8 = 4x \] \[ x = 2 \]
Step 4: Substitute \(x\) back to find \(y\)
Now, substitute \(x = 2\) into either of the original equations. We'll use \(y = -x + 6\):
\[ y = -2 + 6 = 4 \]
Conclusion
Thus, the solution to the system of equations, or the point of intersection, is:
\[ (2, 4) \]
So, the estimated graphically and algebraically determined solution to the system of equations is \( (2, 4) \).