Problem 23 - Algebra 1: Concepts and Skills

Question

Choose a solution method to solve the linear system. Explain your choice, and then solve the system. $$ \begin{array}{r} {2 x-3 y=-7} \\ {3 x+y=-5} \end{array} $$

Step-by-Step Solution

Answer

The solution of the system of equations is \(x = -2\) and \(y = 1\).

Step 1: Multiply the second equation by 3

Multiply every term in the second equation by 3. This will make the y coefficient 3 in the second equation. This gives us: \[ \begin{array}{r} {2 x-3 y=-7} \ {9 x+3 y=-15} \end{array} \]

Step 2: Add the two equations

This will eliminate the y variable. Adding the two equations, we get \(11x = -22\).

Step 3: Solve for x

We can easily solve for x from 11x = -22 by dividing both sides by 11. This gives us \(x = -2\).

Step 4: Substitute x=-2 into the first equation

Substituting x=-2 into the first equation gives \(2*(-2) - 3y = -7\), which simplifies to \(-4 - 3y = -7\). Solve for y by adding 4 to both sides, yielding \(-3y= -3\). Dividing by -3 gives \(y=1\).

Key Concepts

Elimination Method

The elimination method is a technique used to solve a system of linear equations. In this method, we aim to eliminate one variable by adding or subtracting the equations. This allows us to solve for the remaining variable easily.
The primary purpose is to manipulate the equations so that adding or subtracting them will lead to the cancellation of one variable. Here's a streamlined breakdown of the process:

• Identify which variable you want to eliminate.
• Scale one or both equations so that the coefficients of the chosen variable are opposites.
• Add or subtract the equations to eliminate the chosen variable.
• Solve the resulting single-variable equation.
• Finally, substitute back to find the other variable.

The key advantage of the elimination method is that it can simplify the equations significantly when the coefficients are suitable, making the remaining steps straightforward.

Linear Equations

Linear equations are equations that involve two or more variables, where each is raised only to the first power and not multiplied together. They generally have solutions that can be represented by straight lines when graphed. Here are some important characteristics:

• They can have one solution, no solution, or infinitely many solutions.
• The general form of a linear equation in two variables is \( ax + by = c \), where \(a\), \(b\), and \(c\) are constants.
• The solutions to linear equations are the points where the lines intersect if graphed.

The simplicity of linear equations allows for methods like substitution, graphing, or elimination to solve them effectively. Being able to transform these equations through algebraic methods helps solve systems involving several equations.

Algebraic Manipulation

Algebraic manipulation involves rearranging equations to isolate one of the variables. This is crucial in solving systems of equations, as it helps simplify expressions and find solutions. Through algebraic manipulation, equations become more manageable. Key techniques include:

• Combining like terms to simplify the equation.
• Using the distributive property to remove parentheses.
• Adding, subtracting, multiplying, or dividing both sides of the equation to simplify or isolate the variable of interest.

In the original problem, algebraic manipulation was used when multiplying the second equation by 3. This step ensured that the coefficients of \(y\) in both equations were opposites, allowing for the efficient elimination of \(y\). Understanding these manipulative techniques is essential since they form the foundation for solving more complex algebra problems.