Problem 62 - Intermediate Algebra

Question

Explain how you would solve the system $$ \left(\begin{array}{lr} 3 x-4 y= & -1 \\ 2 x-5 y= & 9 \end{array}\right) $$ using the elimination-by-addition method.

Step-by-Step Solution

Answer

The solution to the system is \( \left(\frac{-41}{7}, \frac{-29}{7}\right) \).

Step 1: Align the equations

First, write down the system of equations clearly: \( 3x - 4y = -1 \) and \( 2x - 5y = 9 \). These equations are already properly aligned for elimination.

Step 2: Equalize coefficients of \( x \)

Multiply the first equation by \( 2 \) and the second equation by \( 3 \) to make the coefficients of \( x \) equal:\[ \begin{align*} 2(3x - 4y) &= 2(-1), \ 3(2x - 5y) &= 3(9). \end{align*} \] These simplify to: \( 6x - 8y = -2 \) and \( 6x - 15y = 27 \).

Step 3: Eliminate \( x \) by subtraction

Subtract the second equation from the first to eliminate \( x \):\[ (6x - 8y) - (6x - 15y) = -2 - 27, \] which simplifies to \( 7y = -29 \).

Step 4: Solve for \( y \)

Solve the equation \( 7y = -29 \) by dividing both sides by 7:\[ y = \frac{-29}{7}. \]

Step 5: Substitute \( y \) back to find \( x \)

Substitute \( y = \frac{-29}{7} \) back into the first original equation \( 3x - 4y = -1 \):\[ 3x - 4\left(\frac{-29}{7}\right) = -1. \]Simplify to find \( x \):\[ 3x + \frac{116}{7} = -1. \]Convert \(-1\) to \(\frac{-7}{7}\), then:\[ 3x = \frac{-123}{7}. \]Divide by 3:\[ x = \frac{-123}{21} = \frac{-41}{7}. \]

Step 6: Write the solution as an ordered pair

The solution to the system is \( x = \frac{-41}{7} \) and \( y = \frac{-29}{7} \), so the solution as an ordered pair is \( \left(\frac{-41}{7}, \frac{-29}{7}\right) \).

Key Concepts

System of Linear Equations

A system of linear equations comprises two or more linear equations that share common variables. In our example, the variables are \( x \) and \( y \). Linear equations are equations where each term is either a constant or the product of a constant and a single variable. When combined into a system, these equations work together to pinpoint a specific solution where all the equations in that system intersect.
To visualize this, imagine plotting each equation on a graph. The point where lines intersect is the solution to the system.
In our case, the equations are:

• \( 3x - 4y = -1 \)
• \( 2x - 5y = 9 \)

These are aligned along two axes, and the solution is the point \( \left(\frac{-41}{7}, \frac{-29}{7}\right) \) where the lines intersect.

Solving Equations

Solving equations involves finding the values of the variables that satisfy the equations. For a single equation with two variables, infinite solutions often exist. However, when we introduce a second equation into the mix, like our linear system, these equations work collectively, often resulting in a unique solution.
In solving equations, we apply various operations like addition, subtraction, multiplication, and division to isolate the variables. In this context, we focus on aligning coefficients so the system can be simplified. After aligning or modifying the coefficients, we execute algebraic operations to reduce the number of variables, ultimately solving for the unknown values.
The solution of our system, achieved through elimination-by-addition, is \( x = \frac{-41}{7} \) and \( y = \frac{-29}{7} \).

Algebraic Methods

Algebraic methods offer various strategies to solve systems of equations. The elimination-by-addition method is an effective algebraic strategy. Here's a concise explanation of how it works.

• Align Variables: Ensure equations are written in a format where similar terms are aligned, as done in our example equations: \( 3x - 4y = -1 \) and \( 2x - 5y = 9 \).
• Equalize Coefficients: Modify equations by multiplying with constants to equalize the coefficients of one variable. In our exercise, the \( x \)-coefficients were equalized by multiplying different constants.
• Eliminate a Variable: After alignment, subtract or add equations to eliminate a variable. We subtracted two simplified equations to eliminate \( x \).
• Substitute and Solve: Once one variable is known, substitute it back into any of the original equations to find the other unknown.
• Check Solution: Always verify your results to ensure equations are satisfied by the derived solutions.

By mastering these steps, you'll tackle similar problems with confidence and precision.