Problem 42 - Intermediate Algebra

Question

For Problems 19-48, solve each system by using either the substitution or the elimination-by-addition method, whichever seems more appropriate. (Objective 2) $$ \left(\begin{array}{l} 2(x-1)-3(y+2)=30 \\ 3(x+2)+2(y-1)=-4 \end{array}\right) $$

Step-by-Step Solution

Answer

The solution is \((4, -10)\).

Step 1: Simplify the Equations

First, we expand and simplify both equations to avoid complex expressions.For the first equation: \[ 2(x-1) - 3(y+2) = 30 \]Expanding gives:\[ 2x - 2 - 3y - 6 = 30 \]Simplify to get:\[ 2x - 3y = 38 \]For the second equation:\[ 3(x+2) + 2(y-1) = -4 \]Expanding gives:\[ 3x + 6 + 2y - 2 = -4 \]Simplify to get:\[ 3x + 2y = -8 \]

Step 2: Rearrange Equations into Standard Form

Write the simplified equations in standard form to make it easier for elimination.\[ 2x - 3y = 38 \]\[ 3x + 2y = -8 \]

Step 3: Use Elimination Method

To eliminate using the elimination method, we adjust the coefficients.Multiply the first equation by 2 and the second equation by 3 to align the coefficients of \(y\):First equation:\[ 4x - 6y = 76 \]Second equation:\[ 9x + 6y = -24 \]Add both equations to eliminate \(y\):\[ 4x - 6y + 9x + 6y = 76 - 24 \]This simplifies to:\[ 13x = 52 \]

Step 4: Solve for x

Solve the equation for \(x\):\[ 13x = 52 \]Divide both sides by 13:\[ x = 4 \]

Step 5: Substitute x to Solve for y

Substitute \(x = 4\) back into one of the original equations to solve for \(y\).Using the first simplified equation:\[ 2 \times 4 - 3y = 38 \]\[ 8 - 3y = 38 \]\[ -3y = 30 \]Divide by -3:\[ y = -10 \]

Step 6: Write the Solution

The solution to the system of equations \(x = 4\) and \(y = -10\) satisfies both equations. Therefore, the solution to the system is:\[ (4, -10) \]

Key Concepts

Elimination Method

When solving systems of linear equations, the elimination method helps us find the solution by eliminating one of the variables. The goal is to add or subtract equations so that one variable disappears. This method is effective for equations that are already in or can be easily rearranged into a form where adding or subtracting will cancel out one of the variables.

Here's a simple way to think about it:

• Start by aligning equations so that like terms are directly underneath each other.
• Manipulate the equations by multiplying them with suitable factors to get equal coefficients for one of the variables across the two equations.
• Add or subtract the equations to eliminate one variable, then solve for the remaining variable.

This method is particularly useful when equations are already neat and no messy fractions or decimals are involved, making it a straightforward process to eliminate variables.

Substitution Method

The substitution method is an alternative approach for solving systems of equations. This technique involves solving one equation for one variable, and then substituting that expression into the other equation. It works well when one of the equations easily isolates one variable, making it simpler to substitute.

How to use substitution:

• Choose an equation to solve for one variable in terms of the other (e.g., solve for \( x \) in terms of \( y \)).
• Substitute the expression for the isolated variable into the other equation.
• Solve this new equation for the remaining variable.
• Finally, substitute the found value back into the equation from the first step to find the value of the initial variable.

The substitution method is perfect for smaller systems or when the problem gives a clear choice for substituting, helping to simplify complex calculations.

Linear Equations

Linear equations are equations of the first degree, meaning their highest exponent is 1. They form straight lines when graphed and typically have solutions that can be found using methods like elimination and substitution.

Characteristics of linear equations:

• They form a straight line on a graph.
• They consist of variables with no powers greater than 1 (e.g., \( ax + by = c \)).
• The solutions represent the points where the lines, if graphed, would intersect.

Understanding linear equations sets the stage for solving systems of equations, primarily because of their simplicity and straightforward nature. Recognizing whether an equation is linear makes it easier to decide on the best solving technique.