Problem 24 - Precalculus

Question

Solve each system of linear equations by elimination. $$\begin{array}{r} -2 x+3 y=1 \\ 2 x-y=7 \end{array}$$

Step-by-Step Solution

Answer

The solution is \((x, y) = (5.5, 4)\).

Step 1: Add the Equations

The system of equations is: 1. \(-2x + 3y = 1\)2. \(2x - y = 7\) Add the two equations together to eliminate \(x\):\[-2x + 3y + (2x - y) = 1 + 7\] This simplifies to:\[2y = 8\]

Step 2: Solve for y

Simplify the equation \(2y = 8\) by dividing both sides by 2 to solve for \(y\): \[y = \frac{8}{2} = 4\] Thus, \(y = 4\).

Step 3: Substitute y back into one of the original equations

Substitute \(y = 4\) back into the second original equation \(2x - y = 7\):\[2x - 4 = 7\] Solve for \(x\) by adding 4 to both sides:

Step 4: Solve for x

From \(2x - 4 = 7\), add 4 to both sides to get:\[2x = 11\] Now, divide both sides by 2 to find \(x\):\[x = \frac{11}{2} = 5.5\] Thus, \(x = 5.5\).

Key Concepts

System of Linear Equations

A system of linear equations consists of two or more equations with the same set of variables. In our problem, we have a system with two equations:

• \(-2x + 3y = 1\)
• \(2x - y = 7\)

These equations are called 'linear' because their graphs produce straight lines on a coordinate plane. The goal is to find the values of the variables, in this case, \(x\) and \(y\), that will make both equations true simultaneously.

There are several methods to solve such systems, including graphing, substitution, and elimination. The elimination method is particularly useful when you can add or subtract the equations to remove a variable. This can often simplify finding the solution.

Solving Linear Equations

To solve a system of linear equations, we need to find the variable values that satisfy all the given equations. In this solution, we used the elimination method to find the solutions.

First, we added the two equations in the system:

• \(-2x + 3y = 1\)
• \(2x - y = 7\)

By adding them, the \(x\) terms cancel each other out:\[-2x + 3y + 2x - y = 1 + 7\]

This leaves us with a simpler equation involving only \(y\):
\[2y = 8\]

From here, we can solve for \(y\) by simplifying:
\(y = 4\).

With the value of \(y\) known, we substitute it back into one of the original equations to find \(x\). In this solution, we used the second equation \(2x - y = 7\). After substitution, it simplifies and we find \(x = 5.5\).

This way of solving the system made the process more straightforward and efficient, allowing us to find the values of \(x\) and \(y\) that satisfy both equations.

Substitution Method

The substitution method is another approach to solving systems of linear equations, though it wasn't used in this particular solution. This method involves solving one equation for one variable, and then substituting that expression into the other equation.

Here's a brief rundown of how you might use substitution with the original system:1. Solve one of the equations for \(x\) or \(y\). For example, from the second equation \(2x - y = 7\), solve for \(y\):
\[y = 2x - 7\]2. Substitute \(y = 2x - 7\) back into the first equation \(-2x + 3y = 1\):\[-2x + 3(2x - 7) = 1\]3. This substitution gives you an equation with only \(x\), which you solve to find \(x\).
4. Finally, use the value of \(x\) you found to solve for \(y\) using the substituted equation.

The substitution method is handy when one variable is already isolated or easy to isolate. It can be more intuitive for certain types of systems, particularly when you're solving by hand.