Problem 47 - Algebra 2

Question

Juanita and Jamal are solving the system \(2 x-y=6\) and \(2 x+y=10 .\) Who is correct? Explain your reasoning.

Step-by-Step Solution

Answer

The correct solution is \(x = 4\) and \(y = 2\); the person with these values is correct.

Step 1: Understand the System of Equations

First, identify each equation in the system. We have two equations: \(2x - y = 6\) and \(2x + y = 10\). We need to find a solution for \(x\) and \(y\) that satisfies both equations simultaneously.

Step 2: Add the Equations Together

Notice that the two equations have \(2x\) in common. We can eliminate \(y\) by adding the two equations together: \[(2x - y) + (2x + y) = 6 + 10\]This simplifies to \[4x = 16\].

Step 3: Solve for x

From the equation \(4x = 16\), solve for \(x\) by dividing both sides by 4: \[x = \frac{16}{4} = 4\].

Step 4: Substitute x Back into an Equation

We can substitute \(x = 4\) back into either original equation to solve for \(y\). Let's use the first equation: \[2(4) - y = 6\]which simplifies to \[8 - y = 6\].

Step 5: Solve for y

Now solve the equation \(8 - y = 6\) for \(y\): \[-y = 6 - 8\] which simplifies to \(-y = -2\). Multiply both sides by -1 to get \(y = 2\).

Step 6: Verify the Solution

We found \(x = 4\) and \(y = 2\). Substitute these back into the second equation to verify:\[2(4) + 2 = 8 + 2 = 10\]This holds true, confirming our solution.

Step 7: Conclusion on Who is Correct

Both equations are satisfied with \(x = 4\) and \(y = 2\), so the correct solution is dependent on who among Juanita and Jamal arrived at these values. Both equations are satisfied with the same \(x\) and \(y\).

Key Concepts

Elimination Method

The elimination method is a popular technique for solving systems of equations. It involves eliminating one of the variables by adding or subtracting the equations. This makes it easier to solve for the remaining variable.
In this scenario, we look at the two equations:

• Equation 1: \(2x - y = 6\)
• Equation 2: \(2x + y = 10\)

The strategy is to eliminate the \(y\) variable. We'll add the two equations together. Notice how the \(-y\) from the first equation and the \(+y\) from the second equation will cancel each other out. Performing the addition gives us \(4x = 16\), which directly leads us to solve for \(x\).
By using the elimination method, you can efficiently find the solution of the system as it's very effective when the coefficients of one variable are already aligned.

Substitution Method

The substitution method is a bit different from elimination. It's a technique where you solve one of the equations for one variable and then substitute this expression into the other equation.
To illustrate with our example:

• Choose an equation, let's take \(2x - y = 6\).
• Solve for \(y\): \(y = 2x - 6\).
• Substitute \(y = 2x - 6\) into the second equation: \(2x + (2x - 6) = 10\).

After substitution, you have a single equation with one variable, which is easier to solve. This technique is particularly useful when one of the equations is easily manipulated, as it simplifies the process of finding the solution.

Linear Equations

Linear equations are mathematical statements that describe a line when graphed in a coordinate plane. They are expressed in the standard form \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants.
In our exercise, \(2x - y = 6\) and \(2x + y = 10\) are both linear equations. This means their graphs would be straight lines on a coordinate plane. The solution to these equations is the point where the lines intersect.
Understanding linear equations and how they behave on a graph is foundational for solving systems of equations, as it visually demonstrates where solutions exist, if at all.

Simultaneous Equations

Simultaneous equations are a set of equations with multiple variables that you solve at the same time. The goal is to find a common solution for all equations in the system.
In the given task, solving simultaneously means finding values for \(x\) and \(y\) that make both equations true. Both \(2x - y = 6\) and \(2x + y = 10\) had to be satisfied with the same values of \(x\) and \(y\).
Typically, methods such as elimination or substitution are used to solve these types of equations. Recognizing that solving these equations means determining the point of intersection on their graphical representation can also help in understanding the nature of simultaneous equations.