Verified Answer :

  • We analyze the linear system consisting of two equations with two variables.
  • We apply the elimination method by multiplying the first equation by 5 to align the coefficients of \(x\), which allows us to cancel \(x\) by adding the equations.
  • After finding \(y = -1\), we substitute it back into the first equation to solve for \(x\), yielding \(x = 4\).
  • The solution to the system is \(\boxed{x=4, y=-1}\).

Explanation

  1. Problem Analysis

We are given a system of two linear equations in two variables:
\[x + 3y = 1 \quad \text{(1)}\]
\[-5x + 4y = -24 \quad \text{(2)}\]
Our goal is to find the values of \(x\) and \(y\) that satisfy both equations. A system of linear equations represents two lines in a 2D plane, and the solution is the point where these two lines intersect. We can solve this system using the method of elimination, which involves manipulating the equations to isolate one variable.

  1. Elimination Method

To eliminate the variable \(x\), we look at the coefficients of \(x\) in both equations. In equation (1), the coefficient is \(1\), and in equation (2), the coefficient is \(-5\). If we multiply equation (1) by \(5\), the coefficient of \(x\) becomes \(5\), which is the additive inverse of \(-5\).
Multiplying equation (1) by \(5\):
\[5(x + 3y) = 5(1)\]
\[5x + 15y = 5 \quad \text{(3)}\]
Now, we add equation (3) to equation (2):
\[(5x + 15y) + (-5x + 4y) = 5 + (-24)\]
\[(5x - 5x) + (15y + 4y) = -19\]
\[19y = -19\]
Dividing both sides by \(19\), we find:
\[y = -1\]

  1. Solving for Variables and Verification

Now that we have the value of \(y\), we substitute it back into equation (1) to solve for \(x\):
\[x + 3(-1) = 1\]
\[x - 3 = 1\]
Adding \(3\) to both sides, we get:
\[x = 4\]
To verify the solution, we check these values in the second equation:
\[-5(4) + 4(-1) = -20 - 4 = -24\]
The values satisfy both equations. Therefore, the unique solution to the system is \(x = 4\) and \(y = -1\).

Examples

Linear systems are frequently used in business and economics to find the "equilibrium point". Imagine a company selling a product where the cost to produce is related to the quantity and fixed overheads, and the revenue is determined by the sales price. By setting up equations for both cost and revenue, you can determine the point where the company breaks even. For example, if costs follow \(y = 2x + 10\) and revenue follows \(y = 5x - 5\), finding where these lines intersect helps the business owner understand how many units must be sold to cover costs.

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