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Chapter 4: Problem 18

Solve each system by the elimination method. $$ \begin{aligned} &5 x+4 y=12\\\ &3 x+5 y=15 \end{aligned} $$

### Short Answer

Expert verified

x = 0, y = 3

## Step by step solution

01

## Align the equations

Write the given system of equations neatly one below the other. \[5x + 4y = 12 \ 3x + 5y = 15\]

02

## Eliminate one variable

To eliminate one variable, multiply the equations by appropriate constants. Here, multiply the first equation by 3 and the second equation by 5 to make the coefficients of x equal:\[ \ 3(5x + 4y = 12) \ \rightarrow 15x + 12y = 36 \ 5(3x + 5y = 15) \ \rightarrow 15x + 25y = 75 \]

03

## Subtract the equations

Subtract the second equation from the first to eliminate x: \[ (15x + 12y) - (15x + 25y) = 36 - 75 \ 0x - 13y = -39 \ -13y = -39 \]

04

## Solve for y

Solve the resulting equation for y: \[ y = \frac{39}{13} \ y = 3 \]

05

## Substitute y back into one of the original equations

Substitute y = 3 into one of the original equations to find x. Using the first equation: \[ 5x + 4(3) = 12 \ 5x + 12 = 12 \ 5x = 0 \ x = 0 \]

06

## State the solution

The solution to the system of equations is: \[ x = 0, y = 3 \]

## Key Concepts

These are the key concepts you need to understand to accurately answer the question.

###### solve systems of equations

Solving systems of equations involves finding values for the variables that make all equations true simultaneously. In other words, you're looking for where the equations intersect when plotted on a graph. There are several methods to solve systems of equations, including graphing, substitution, and elimination. Each method has its own advantages, but in this article, we'll focus on the elimination method, which is particularly useful when both equations are in standard form.

###### linear equations

Linear equations are equations that produce straight lines when plotted on a graph. They typically follow the format: \[ Ax + By = C \] where \(A\), \(B\), and \(C\) are constants. If you have two linear equations, they can either intersect at a single point, be parallel (hence, no solution), or be the same line (infinitely many solutions). Understanding this basic concept helps in visualizing and solving systems of equations effectively. Let’s take a close look at our given example to make things clearer: \[ \begin{aligned} &5x+4y=12\ &3x+5y=15 \end{aligned} \]

###### elimination method steps

The elimination method involves eliminating one of the variables by combining the equations. Here's a clear breakdown of the steps to solve the given system using this method:Step 1: Align the equationsMake sure both equations are lined up neatly. This helps in easy visualization and calculation: \[ \begin{aligned} &5x+4y=12\ &3x+5y=15 \end{aligned} \]Step 2: Make the coefficients of one variable the sameWe aim to eliminate one variable by making the coefficient of \(x\) the same in both equations. Multiply the first equation by 3 and the second by 5: \[ \begin{aligned} &3(5x+4y) = 3(12) \ &5(3x+5y) = 5(15) \end{aligned} \]This transforms our system into: \[ \begin{aligned} &15x + 12y = 36 \ &15x + 25y = 75 \end{aligned} \]Step 3: Eliminate one variableSubtract the second equation from the first to eliminate \(x\): \[ (15x + 12y) - (15x + 25y) = 36 - 75 \ 0x - 13y = -39 \]Simplifying gives us: \[ -13y = -39 \ y = \frac{39}{13} \ y = 3 \]Step 4: Substitute back to find the other variableNow, substitute \(y=3\) back into one of the original equations to solve for \(x\): \[ 5x + 4(3) = 12 \ 5x + 12 = 12 \ 5x = 0 \ x = 0 \]Step 5: State the solutionThe solution for the system of equations is \(x=0\) and \(y=3\). So, the intersection point is \((0, 3)\).Using these structured steps ensures you can solve any system of equations using the elimination method efficiently.

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