Problem 31 - Precalculus

Question

Solve each system of linear equations by elimination. $$\begin{aligned} &3 x-2 y=12\\\ &4 x+3 y=16 \end{aligned}$$

Step-by-Step Solution

Answer

The solution is \(x = 4\) and \(y = 0\).

Step 1: Identify and Align Coefficients

To use the elimination method, the first step is to make sure that either the coefficients of \(x\) or \(y\) from both equations are the same (or can be easily made the same). Let's aim to eliminate \(y\). In this case, the coefficients of \(y\) are \(-2\) and \(3\), so we will multiply the first equation by \(3\) and the second equation by \(2\) to make the coefficients of \(y\) equal (but opposite in sign).

Step 2: Multiply for Elimination

Multiply both equations as decided:First Equation: \(3(3x - 2y) = 3(12)\) which simplifies to: \(9x - 6y = 36\)Second Equation: \(2(4x + 3y) = 2(16)\) which simplifies to: \(8x + 6y = 32\)

Step 3: Add the Equations

Now, add the two equations together to eliminate \(y\):\[(9x - 6y) + (8x + 6y) = 36 + 32\] The \(-6y\) and \(+6y\) cancel each other out, so we have:\(17x = 68\)

Step 4: Solve for x

To find the value of \(x\), divide both sides of the equation \(17x = 68\) by \(17\):\(x = \frac{68}{17} = 4\)

Step 5: Substitute to Find y

Substitute \(x = 4\) back into either original equation to solve for \(y\). Let's use the first equation:\(3(4) - 2y = 12\) which simplifies to:\(12 - 2y = 12\)Now solve for \(y\): \(-2y = 12 - 12 \Rightarrow -2y = 0\) \(y = \frac{0}{-2} = 0\)

Step 6: Verify the Solution

It's important to verify the solution by plugging \(x = 4\) and \(y = 0\) back into the second equation:\(4(4) + 3(0) = 16\) which simplifies to:\(16 = 16\)Since this holds true, the solution is verified.

Key Concepts

Elimination Method

The elimination method is an effective approach for solving systems of linear equations. It involves strategically manipulating the equations to cancel out one of the variables, simplifying the process of finding a solution. To begin, look at the coefficients of each variable in two equations. The goal is to make one pair of coefficients the same so that adding or subtracting the equations will eliminate that variable.

In our example, we start by aiming to eliminate the variable \(y\). The coefficients of \(y\) in the first and second equations are \(-2\) and \(3\) respectively. To prepare these for elimination, multiply the entire first equation by \(3\) and the entire second equation by \(2\). This results in the coefficients of \(y\) becoming \(-6\) and \(6\), making them additive inverses.

• Multiply the equations to get identical (but opposite) coefficients for one variable.
• Add or subtract the equations to eliminate one variable.
• Solve for the remaining variable first.

By applying this technique, we reduce the complexity of solving simultaneous equations, paving the way to find the solutions for the variables.

Linear Combination

Linear combination is the mathematical essence of the elimination method. It involves adding or subtracting entire equations to simplify and solve them. Essentially, a linear combination involves creating new equations by adding or subtracting multiples of original equations, making it easier to isolate variables.

In solving the given equations using this method, we multiplied each equation by a specific number to prepare for elimination. Specifically, we derive:

• First Equation: \(9x - 6y = 36\)
• Second Equation: \(8x + 6y = 32\)

By adding these modified equations, \(-6y\) and \(+6y\) cancel each other out, leaving only the \(x\) variable. This simplifies the system into one equation with one variable: \(17x = 68\).

This linear combination results in a simpler scenario that allows for straightforward solving of the variable, thus making the entire system manageable and less prone to errors.

Solving for Variables

Once you have managed to eliminate one variable, the next step is solving for the other. This process involves straightforward algebraic manipulation to isolate the variable. Using the simplified equation from our example: \(17x = 68\), divide both sides by \(17\) to solve for \(x\):

• \(x = \frac{68}{17}\)
• \(x = 4\)

With the value of \(x\) found, substitute back into any original equation to find \(y\). Let's substitute into \(3x - 2y = 12\):

\(3(4) - 2y = 12\) simplifies to \(12 - 2y = 12\).

• Simplify to find \(y\): \(-2y = 0\)
• \(y = 0\)

Finally, verify both values by substituting \(x = 4\) and \(y = 0\) into the other original equation, ensuring it holds true. This verification guarantees the solution is correct, confirming both the process and accuracy of your work.