To estimate the solution to the system of the equations given:
- \( y = -x + 6 \) (Line 1)
- \( y = 3x - 2 \) (Line 2)
you can visualize these equations as lines on a Cartesian plane.
Steps to Estimate the Solution:
-
Graph the first equation:
- The y-intercept is 6 (point (0, 6)).
- The slope is -1, which means the line will go down one unit for every unit it goes to the right. You can find another point by moving 1 unit to the right (to (1, 5)) and then down 1 unit.
-
Graph the second equation:
- The y-intercept is -2 (point (0, -2)).
- The slope is 3, meaning for every unit you move to the right, you go up 3 units. From the y-intercept (0, -2), you can go 1 unit right to (1, 1) (up 3 units).
Finding the Intersection:
Once both lines are graphed, you will be looking for the point where they intersect. This intersection point is the solution to the system of equations.
Estimating the Intersection Point:
Graphically estimating from a plot:
- It may be beneficial to use a graphing tool or plot on graph paper to see the exact point.
- Alternatively, you can set the equations equal to find the precise coordinates algebraically.
Algebraic Method:
Set the two equations equal to each other: \[ -x + 6 = 3x - 2 \]
Solving for \( x \): \[ 6 + 2 = 3x + x \] \[ 8 = 4x \] \[ x = 2 \]
Now, substitute \( x = 2 \) back into either equation to find \( y \): Using the first equation: \[ y = -2 + 6 = 4 \]
Solution:
The estimated solution to the system of equations is the point \( (2, 4) \). This means the two lines intersect at this point.