Answer:
Explanation:
The solution to two, nonparallel linear equations is the point at which they intersect. We know they will intersect if 1) they are equations of straight lines, and 2) their slopes are different.
Lets rewrite the 2nd equation:
3x − y = 6
−y = -3x + 6
y = 3x - 6
We want the point of intersection of these two equations. We can see that they are linear and that the two slopes (-1 and 3) are not the same. They will intersect.
y = −x + 2
y = 3x - 6
We can solve this by one of two methods: Graphing or mathematical. Lets do both.
Graphing
See the attached graph (DESMOS). The lines intersect at (2,0). This is a fast and easy solution, if you have access to a free graphing utility such as DESMOS, but it sometimes requires careful analysis of the intersection point if it does not fall exactly on the axis markers.
Mathematically
Since we want the point at which both equations have the same values of x and y, lets set the y values (y1 and y2) equal to each other and find the value of x that make the equations equal.
1) y1 = −x + 2
2) y2 = 3x - 6
Set y1 = y2:
−x + 2 = 3x - 6
-4x = -8
x = 2
Use this value of x to find y:
y1 = −x + 2
y1 = −2 + 2
y1 = 0
CHECK, using the second equation:
y2 = 3x - 6
y2 = 3*(2) - 6
y2 = 6-6
y2 = 0 As expected, and now we have the solution:
The 2 lines intersect at (2,0).
Graphing to find the solution is easy, but the mathematical approach is more satisfying (for me). Do both, since finding the same answer serves as a good check on the solution.
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