Problem 14 - Elementary Algebra

Question

Use the elimination-by-addition method to solve each system. $$\left(\begin{array}{l}2 x-7 y=-2 \\ 3 x+y=1\end{array}\right)$$

Step-by-Step Solution

Answer

The solution is \(x = \frac{5}{23}\) and \(y = \frac{8}{23}\).

Step 1: Align the System of Equations

Given the system of equations: 1) \(2x - 7y = -2\) 2) \(3x + y = 1\).Let's align them: \( \begin{cases} 2x - 7y = -2 \ 3x + y = 1 \end{cases} \). We need to manipulate these equations to eliminate one of the variables.

Step 2: Eliminate y by Addition

Next, we multiply the second equation by 7 to align the coefficients of \(y\), so:\(3x + y = 1\) becomes \(21x + 7y = 7\). Now, the system is:\( \begin{cases} 2x - 7y = -2 \ 21x + 7y = 7 \end{cases} \).Adding these two equations eliminates \(y\): \( (2x - 7y) + (21x + 7y) = -2 + 7 \).

Step 3: Solve for x

After adding the equations, we have:\(23x = 5\).Solve for \(x\) by dividing both sides by 23:\(x = \frac{5}{23}\).

Step 4: Substitute x to Find y

Substitute \(x = \frac{5}{23}\) back into one of the original equations, preferably the simpler one, \(3x + y = 1\). Thus, \(3(\frac{5}{23}) + y = 1\) gives \(\frac{15}{23} + y = 1\).Solve for \(y\) by subtracting \(\frac{15}{23}\) from 1:\(y = 1 - \frac{15}{23} = \frac{23}{23} - \frac{15}{23} = \frac{8}{23}\).

Step 5: Verify the Solution

Substitute \(x = \frac{5}{23}\) and \(y = \frac{8}{23}\) into the other equation, \(2x - 7y = -2\), to verify:\(2(\frac{5}{23}) - 7(\frac{8}{23}) = \frac{10}{23} - \frac{56}{23} = -\frac{46}{23} = -2\), so the solution satisfies this equation.

Key Concepts

Understanding Systems of Equations

In mathematics, a **system of equations** refers to a set of two or more equations with the same variables. Solving these systems aims to find the values of the variables that satisfy all the equations simultaneously. Consider the provided pair of linear equations:

• Equation 1: \(2x - 7y = -2\)
• Equation 2: \(3x + y = 1\)

These equations represent a system because they involve the same variables, \(x\) and \(y\). Solving the system provides a value for each variable that works for both equations at the same time. Visualize this as two lines on a graph; the solution is where these lines intersect. In this method, our goal is to simplify both equations to make finding this intersection straightforward.

Basics of Solving Linear Equations

A linear equation is an equation of the first degree, meaning its variables are not raised to any powers other than one. The linear equations we are dealing with here have two variables, \(x\) and \(y\). The solution requires finding the exact values of these variables that satisfy both equations. Solving methods include:

• Substitution Method: Solve one equation for one variable, then substitute in the other equation.
• Graphical Method: Graph the equations to find the intersection point signifying the solution.
• Elimination Method: Manipulate the equations to eliminate one variable, making it easier to solve for the remaining variable.

In our exercise, we choose the elimination method as it efficiently addresses the problem given the setup of the equations. By manipulating the terms in these equations, particularly via multiplication and addition, we can cancel one of the variables to solve the equations step-by-step.

Algebraic Manipulation through Elimination-by-Addition Method

The elimination-by-addition method is a popular algebraic technique used to solve systems of linear equations. Its goal is to remove one variable by adding or subtracting equations from each other. Here's how we use it:1. **Align the Equations:** Start by writing both equations clearly. For example, given our system: - \(2x - 7y = -2\) - \(3x + y = 1\)2. **Eliminate a Variable:** Choose a variable to eliminate. In this case, we targeted \(y\) due to its neat coefficients. We multiplied the entire second equation by 7 to align the coefficients of \(y\), resulting in: - \(21x + 7y = 7\) 3. **Addition to Cancel out y:** Add the modified second equation to the first: - \[(2x - 7y) + (21x + 7y) = -2 + 7\] - This simplifies the equations and cancels out \(y\).4. **Solve for Remaining Variable:** Simplify to solve for \(x\), making calculations straightforward to handle.5. **Substitute back and Solve for Other Variable:** Use the value found for \(x\) and substitute it back into one of the original equations to find \(y\).This technique requires good algebraic manipulation skills but simplifies the complex system into single-variable problems that are much easier to solve.