Problem 13 - Precalculus with Limits

Question

Solving a System by Elimination In Exercises \(13-30\) , solve the system by the method of elimination and check any solutions algebraically. $$ \left\\{\begin{array}{l}{x+2 y=6} \\ {x-2 y=2}\end{array}\right. $$

Step-by-Step Solution

Answer

The solution to the system of equations is \(x = 4\) and \(y = 1\).

Step 1: Add the Equations

Add the two equations to eliminate the \(y\) term. This is done like this: \( (x + 2y) + (x - 2y) = 6 + 2 \). This simplifies to \(2x = 8\).

Step 2: Solve for the remaining variable

Now, solve the resulting equation for the remaining variable \(x\), \( 2x = 8 \) divides both sides by 2, then \(x = 4\).

Step 3: Substitution

Substitute the obtained value of \(x\) into one of our original equations to solve for \(y\). If we substitute \(x = 4\) into \(x + 2y = 6\), we get \(4 + 2y = 6\), which simplifies to \(2y = 2\).

Step 4: Solve for the second variable

We solve the equation \(2y = 2\) for \(y\). When we divide both sides by 2, we find that \(y = 1\).

Step 5: Check solution

Substitute the values of \(x = 4\) and \(y = 1\) into the original equations to check if they hold true. Substituting these values into both equations we get \( 4 + 2*1 = 6\) and \( 4 - 2*1 = 2 \), both equations hold true, therefore the solution is correct.

Key Concepts

Elimination Method

The elimination method is a powerful technique to solve a system of linear equations. Instead of dealing with two variables at once, we aim to eliminate one variable by adding or subtracting the equations. This simplification makes it easier to solve for the remaining variable.

In our example setup, we have two equations: \(x + 2y = 6\) and \(x - 2y = 2\). Notice the coefficients of \(y\) are \(+2\) and \(-2\). This is the perfect setup for elimination. By adding these equations, the \(y\) terms cancel each other out:

• Add: \((x + 2y) + (x - 2y) = 6 + 2\)
• Result: \(2x = 8\)

Now, we're left with a single-variable equation, easy to solve. Remember, the key to this method is getting one variable to cancel out, simplifying the problem down to one equation.

Algebraic Solutions

Algebraic solutions involve finding the exact answer to an equation or set of equations. In this case, we start by solving for \(x\) once we have simplified the system using elimination. With \(2x = 8\):

• Divide both sides by 2: \(x = 4\)

This tells us the precise value of \(x\).

Algebraic solutions are about precision. We follow steps logically to arrive at the answer without guessing. Verification of the solution is also a part of this process to ensure the values satisfy all original equations. It's like solving a puzzle, where each piece must fit perfectly.

Variables Substitution

Once we have determined one variable using the elimination method, substitution comes into play. Here, we replace the solved variable back into one of the original equations to find the other variable.

Let's substitute \(x = 4\) back into the equation \(x + 2y = 6\):

• We get: \(4 + 2y = 6\)
• Solving for \(y\): Subtract 4 from both sides to get \(2y = 2\)
• Divide by 2: \(y = 1\)

Substitution helps us complete the solution for all variables involved. By replacing known values, we can precisely calculate the unknowns. This step is essential in confirming that our work with elimination has provided correct results. Verifying through substitution ensures the entire solution holds true across both original equations.