Verified Answer :

  • Identify slopes and y-intercepts for both lines: \(y = -3x - 2\) (slope -3, intercept -2) and \(y = x + 6\) (slope 1, intercept 6).
  • Find two points for each line to plot them: \((0, -2), (-1, 1)\) for the first and \((0, 6), (-2, 4)\) for the second.
  • Graph both lines on a coordinate plane and observe the point of intersection.
  • Confirm the intersection point \((-2, 4)\) by substituting the values into the original equations.
  • The solution to the system is \(\boxed{(-2, 4)}\).

Explanation

  1. Problem Analysis and Identifying Key Features

We are given a system of two linear equations in two variables:
1) \(y = -3x - 2\)
2) \(y = x + 6\)
To solve this system by graphing, we interpret each equation as a line on a Cartesian plane. A solution to the system corresponds to the coordinates \((x, y)\) where the two lines intersect. Both equations are already in the slope-intercept form \(y = mx + b\), where \(m\) is the slope and \((0, b)\) is the y-intercept. For the first line, the slope \(m_1 = -3\) and the y-intercept is \((0, -2)\). For the second line, the slope \(m_2 = 1\) and the y-intercept is \((0, 6)\).

  1. Finding Points for Plotting

To graph the first line \(y = -3x - 2\), we can use the y-intercept \((0, -2)\). To find another point, let \(x = -1\):
\(y = -3(-1) - 2 = 3 - 2 = 1\).
So, the line passes through \((0, -2)\) and \((-1, 1)\).
To graph the second line \(y = x + 6\), we use the y-intercept \((0, 6)\). To find another point, let \(x = -2\):
\(y = -2 + 6 = 4\).
So, the line passes through \((0, 6)\) and \((-2, 4)\).

  1. Graphing and Identifying the Intersection

Now, plot these points on a coordinate system. For the first line, draw a straight line through \((0, -2)\) and \((-1, 1)\). For the second line, draw a straight line through \((0, 6)\) and \((-2, 4)\). By observing the graph, we see that the two lines intersect at the point \((-2, 4)\). This means that \(x = -2\) and \(y = 4\) should be the solution for the system.

  1. Verification and Conclusion

To ensure our result is accurate, we verify the point \((-2, 4)\) by substituting it into both original equations:
1) For \(y = -3x - 2\):
\(4 = -3(-2) - 2 = 6 - 2 = 4\) (Correct)
2) For \(y = x + 6\):
\(4 = -2 + 6 = 4\) (Correct)
Since the point satisfies both equations, it is the correct solution to the system. The intersection point is \((-2, 4)\).

Examples

Graphing systems of linear equations is essential in fields like urban planning and economics. For example, if you are modeling the cost of two different delivery services where one has a flat fee and the other has a per-mile rate, you can graph the total cost functions. The point where the lines intersect represents the distance at which both services cost exactly the same, helping you decide which provider is more economical depending on your delivery needs.

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