Verified Answer :

  • Express the first equation \(x - y = -1\) as \(y = x + 1\).
  • Express the second equation \(x + 2y = 11\) as \(y = -0.5x + 5.5\).
  • Graph both lines on a coordinate system and observe their intersection.
  • The two lines intersect at \((x, y) = \boxed{(3, 4)}\).

Explanation

  1. Understanding the Problem

We are given a system of two linear equations in two variables, \(x\) and \(y\):
1) \(x - y = -1\)
2) \(x + 2y = 11\)
A system of linear equations represents lines on a coordinate plane. The solution to the system is the point \((x, y)\) where the two lines intersect. Our objective is to find this intersection point. To do this, we can express each equation in slope-intercept form, \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. This will make it easy to plot the lines and identify their intersection.

  1. Analyzing the First Equation

First, let's rearrange the equation \(x - y = -1\):
Subtracting \(x\) from both sides gives: \(-y = -x - 1\).
Multiplying by \(-1\) results in: \(y = x + 1\).
This line has a slope \(m_1 = 1\) and a y-intercept \(b_1 = 1\). To graph this, we can pick two points:

  • If \(x = 0\), then \(y = 1\), giving the point \((0, 1)\).
  • If \(x = 1\), then \(y = 2\), giving the point \((1, 2)\).
  1. Analyzing the Second Equation

Next, let's rearrange the equation \(x + 2y = 11\):
Subtracting \(x\) from both sides gives: \(2y = -x + 11\).
Dividing by \(2\) gives: \(y = -\frac{1}{2}x + \frac{11}{2}\) or \(y = -0.5x + 5.5\).
This line has a slope \(m_2 = -0.5\) and a y-intercept \(b_2 = 5.5\). To graph this, we can pick two points:

  • If \(x = 1\), then \(y = -0.5(1) + 5.5 = 5\), giving the point \((1, 5)\).
  • If \(x = 3\), then \(y = -0.5(3) + 5.5 = -1.5 + 5.5 = 4\), giving the point \((3, 4)\).
  1. Finding the Intersection Point

By graphing these two lines, we observe where they meet. The first line passes through \((0, 1)\) and \((1, 2)\). The second line passes through \((1, 5)\) and \((3, 4)\). Notice that the point \((3, 4)\) is on both lines.
Let's verify this algebraically:
For \(x - y = -1\): \(3 - 4 = -1\). (Correct)
For \(x + 2y = 11\): \(3 + 2(4) = 3 + 8 = 11\). (Correct)
The lines intersect at the point \((3, 4)\).

Examples

Solving systems of linear equations is essential in economics, particularly in finding the equilibrium point where supply and demand meet. Imagine a company selling a specific gadget: the demand curve shows how many items consumers buy at different prices, while the supply curve shows how many items the company produces at those prices. By graphing both lines and finding their intersection, the company can determine the optimal price and quantity to balance the market effectively.

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