Problem 4 - Algebra A Combined Function

Question

Solve each system of equations by the addition method. $$ \left\\{\begin{array}{l} x-2 y=-11 \\ -x+5 y=23 \end{array}\right. $$

Step-by-Step Solution

Answer

The solution is \( x = -3 \), \( y = 4 \).

Step 1: Write the System of Equations

The given system of equations is: 1) \( x - 2y = -11 \) 2) \( -x + 5y = 23 \).

Step 2: Add the Equations

Notice that the equations are suitable for addition as adding them eliminates \(x\). Add equation (1): \( x - 2y = -11 \) to equation (2): \( -x + 5y = 23 \). This results in: \[ (x - 2y) + (-x + 5y) = -11 + 23 \] which simplifies to: \[ 3y = 12 \].

Step 3: Solve for y

From the equation \( 3y = 12 \), we divide both sides by 3 to isolate \( y \):\[ y = \frac{12}{3} = 4 \].Thus, \( y = 4 \).

Step 4: Substitute y into One Equation

Substitute \( y = 4 \) back into one of the original equations to solve for \( x \). Let's use equation (1): \( x - 2(4) = -11 \). Simplifying this, we have: \( x - 8 = -11 \).

Step 5: Solve for x

Add 8 to both sides of the equation \( x - 8 = -11 \) to solve for \( x \):\[ x = -11 + 8 = -3 \].Thus, \( x = -3 \).

Step 6: Verify the Solution

Substitute \( x = -3 \) and \( y = 4 \) into the second equation to verify: \( -(-3) + 5(4) = 23 \). Calculate: \( 3 + 20 = 23 \). The equation holds true, confirming our solution.

Key Concepts

Addition Method

The addition method is a strategic way to solve systems of equations. It takes advantage of combining equations to eliminate one of the variables, simplifying the process of finding solutions for both. By aligning terms and adding equations, we aim to cancel out one variable, leaving a simpler equation with just one unknown.

Imagine you have two equations with two variables, like this:

• Equation 1: \( x - 2y = -11 \)
• Equation 2: \( -x + 5y = 23 \)

Notice that if you add these equations, \( x \) and \( -x \) cancel out completely. This is what makes the addition method so useful; it reduces the complexity of the system.

In this scenario, the result of adding these two equations is \( 3y = 12 \), leading directly to a simple solution for \( y \). This method is both time-saving and reduces possible errors because you eliminate a variable early in the process.

Linear Equations

Linear equations are mathematical expressions that create straight lines when graphed on a coordinate plane. They are characterized by variables raised only to the first power and can typically be written in the form \( ax + by = c \).

In the context of solving systems of equations, these properties of linear equations are quite advantageous. Such systems are formed by multiple linear equations, like:

• \( x - 2y = -11 \)
• \( -x + 5y = 23 \)

These are particularly interesting because every solution corresponds to a point where the two lines intersect, indicating the pair \( (x, y) \) that satisfies both equations.

This exercise shows how two seemingly complex equations work together. They reveal simple and tangible solutions when systematically addressed with methods like addition or substitution.

Substitution

Substitution is another method for solving systems of equations and might sometimes follow the addition method when finding the values for variables. This process involves solving for one variable in terms of another and then substituting back into any original equation.

In this exercise, once \( y \) is determined as 4 using the addition method, substitution comes into play to find \( x \). You substitute \( y = 4 \) into one of the initial linear equations:

• \( x - 2(4) = -11 \)

This substitution transforms the problem into a simple equation with one variable, \( x - 8 = -11 \).

Solving for \( x \) gives \( x = -3 \). Using substitution not only simplifies the task but also provides a clear path to finding both variables' values in the system, effectively concluding the search for the complete solution. This method is often used in tandem with others to reach a solution efficiently.