Problem 52 - Algebra A Combined Function

Question

Suppose you are solving the system \(\left\\{\begin{array}{l}-2 x-y=0 \\ -2 x+3 y=6\end{array}\right.\) You decide to use the addition method by multiplying both sides of the first equation by \(3,\) then adding the resulting equation to the second equation. Which of the following is the correct sum? Explain. a. \(-8 x=6\) b. \(-8 x=9\)

Step-by-Step Solution

Answer

The correct sum is \(-8x = 6\).

Step 1: Multiply the First Equation by 3

Start with the original system of equations: \(-2x - y = 0\) and \(-2x + 3y = 6\). Multiply the entire first equation by 3 to facilitate elimination of \(y\). This results in the equation: \(-6x - 3y = 0\).

Step 2: Write Down the Second Equation

Take the second equation as it is, which is: \(-2x + 3y = 6\).

Step 3: Add the Two Equations Together

Add the modified first equation \(-6x - 3y = 0\) to the second equation \(-2x + 3y = 6\). This results in the elimination of \(y\): \(-6x - 3y + (-2x) + 3y = 0 + 6\). Simplify this to get \(-8x = 6\).

Step 4: Identify the Correct Option

After simplifying, you find \(-8x = 6\), which matches option (a) as the original addition result.

Key Concepts

Systems of Equations

A system of equations consists of two or more equations with the same set of variables. In this exercise, we're working with two equations that involve the variables \(x\) and \(y\). Solving a system of equations means finding the values of these variables that satisfy all the equations simultaneously.

There are different ways to solve systems of equations:

• Graphically: Drawing the equations on a graph and identifying the intersection points.
• Substitution: Solving one equation for a variable and substituting that into another equation.
• Elimination: Also known as the addition method, it involves adding or subtracting equations to eliminate a variable.

For this problem, the elimination method was used. When you eliminate a variable, you progressively simplify the system to make it easier to find a solution.

Elimination Method

The elimination method is a strategic way to solve a system of equations by removing one of the variables. This is helpful when one equation can be manipulated to simplify both.

Here’s how the process works in general steps:

• Multiply each equation by a suitable number so that the coefficients of one of the variables are opposite numbers. In this exercise, the first equation was multiplied by 3 to align with the second equation.
• Add or subtract the equations to eliminate one variable. The result should be a new equation with just one variable. Here, adding the equations after the multiplication was key to eliminating \(y\).
• Solve the resulting single-variable equation to find the value of one variable.
• Use the found value in one of the original equations to solve for the other variable.

The elimination makes it clear which variable to focus on and allows you to methodically solve the system without juggling fractions or complicated expressions.

Linear Equations

Linear equations are algebraic expressions where each term is either a constant or the product of a constant and a single variable. These equations can be written in the general form \(ax + by = c\).

In the given system:

• The first equation is \(-2x - y = 0\).
• The second equation is \(-2x + 3y = 6\).

Both equations take on this linear form, allowing them to be manipulated with techniques like the elimination method.

Linear equations are crucial because they form straight lines when graphed. They are foundational in algebra and are essential for understanding relationships between variables. By solving these equations simultaneously, you determine the precise point where both lines intersect. This point represents the solution to the system if a unique solution exists.