Verified Answer :

  • Identify the lines \(y = -3x + 4\) and \(y = 3x - 2\) by their slopes and intercepts.
  • Plot key points such as \((0, 4)\) and \((1, 1)\) for the first line, and \((0, -2)\) and \((1, 1)\) for the second line.
  • Observe the intersection point where both lines share the same coordinates on the Cartesian plane.
  • Confirm the intersection \((1, 1)\) algebraically by substituting into both equations, concluding the final answer is \(\boxed{(1, 1)}\).

Explanation

  1. Analyzing the System of Equations

The problem presents a system of two linear equations in two variables:
1) \(y = -3x + 4\)
2) \(y = 3x - 2\)

Each of these equations represents a straight line on a Cartesian coordinate plane. Our objective is to find the point \((x, y)\) where these two lines intersect. This point, if it exists, is the unique pair of values that satisfies both equations simultaneously. To solve this, we can analyze the lines' characteristics: slopes and y-intercepts. Line 1 has a slope of \(-3\) and a y-intercept at \((0, 4)\). Line 2 has a slope of \(3\) and a y-intercept at \((0, -2)\). Since their slopes are different (\(-3 eq 3\)), the lines must intersect at exactly one point.

  1. Graphing and Finding the Intersection

To solve the system, we visualize both lines on the coordinate plane.

For the first equation, \(y = -3x + 4\), we can identify two points:

  • If \(x = 0\), then \(y = 4\), giving the point \((0, 4)\).
  • If \(x = 1\), then \(y = -3(1) + 4 = 1\), giving the point \((1, 1)\).

For the second equation, \(y = 3x - 2\), we can identify two points:

  • If \(x = 0\), then \(y = -2\), giving the point \((0, -2)\).
  • If \(x = 1\), then \(y = 3(1) - 2 = 1\), giving the point \((1, 1)\).

By plotting these points and drawing the lines through them, we observe that both lines pass through the shared coordinate \((1, 1)\).

  1. Verification and Conclusion

To confirm the intersection point \((1, 1)\), we substitute these values back into the original equations:
1) For \(y = -3x + 4\):
\(1 = -3(1) + 4\)
\(1 = -3 + 4\)
\(1 = 1\) (True)

2) For \(y = 3x - 2\):
\(1 = 3(1) - 2\)
\(1 = 3 - 2\)
\(1 = 1\) (True)

Both equations are satisfied at \((1, 1)\). Therefore, the solution to the system of equations is the point \((1, 1)\), or \(x = 1, y = 1\). The final answer is \(\boxed{(1, 1)}\).

Examples

Systems of linear equations are frequently used in economics and logistics to determine the break-even point. Imagine a business where one equation represents the total cost of production (dependent on the number of units) and the other represents the total revenue from sales. The point where these two lines intersect on a graph is the break-even point, where the company neither makes a profit nor incurs a loss. Finding this intersection helps managers make informed decisions about pricing and production volume to ensure the business remains sustainable.

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