Problem 11 - A Graphical Approach to Precalculus with Limits

Question

Solve each system by substitution. $$\begin{aligned}&3 x-2 y=12\\\&5 x=4-2 y\end{aligned}$$

Step-by-Step Solution

Answer

The solution is \( x = 2 \) and \( y = -3 \).

Step 1: Solve one equation for one variable

First, let's solve the second equation for one of the variables. The second equation is: \( 5x = 4 - 2y \). Solve for \( x \) by isolating it on one side: \( x = \frac{4 - 2y}{5} \).

Step 2: Substitute into the first equation

Now that we have \( x = \frac{4 - 2y}{5} \), substitute this expression for \( x \) into the first equation: \( 3x - 2y = 12 \). This becomes: \( 3\left(\frac{4 - 2y}{5}\right) - 2y = 12 \).

Step 3: Simplify the equation

Distribute the 3 into the expression: \( \frac{12 - 6y}{5} - 2y = 12 \). Next, eliminate the fraction by multiplying every term by 5: \( 12 - 6y - 10y = 60 \).

Step 4: Solve for y

Combine like terms: \( 12 - 16y = 60 \). Solve for \( y \) by subtracting 12 from both sides: \( -16y = 48 \). Divide both sides by \(-16\) to find \( y \): \( y = -3 \).

Step 5: Substitute y back to find x

Now that we know \( y = -3 \), substitute this value back into the expression for \( x \): \( x = \frac{4 - 2(-3)}{5} \). Simplify to find \( x \): \( x = \frac{4 + 6}{5} = \frac{10}{5} = 2 \).

Step 6: Solution verification

Verify the solution by substituting \( x = 2 \) and \( y = -3 \) into the original equations: \( 3(2) - 2(-3) = 12 \) and \( 5(2) = 4 - 2(-3) \). Both are true: \( 6 + 6 = 12 \) and \( 10 = 10 \).

Key Concepts

Substitution Method

The substitution method is a handy tool for solving systems of equations. It involves replacing one variable with an expression containing the other variable. By doing this, we can simplify the equations and solve for one variable at a time. Here's how it works:

• Choose one equation and solve for one of its variables.
• Substitute this solved value into the other equation in place of the variable.
• This substitution results in an equation with just one variable, making it much easier to solve.

This method is especially useful for systems where one equation is already solved for one variable, or can easily be rearranged as such. This was done in the original exercise by expressing \( x \) in terms of \( y \) in the equation \( 5x = 4 - 2y \).
This rearrangement allows us to substitute into the other equation, leading to a straightforward solution.

Linear Equations

Linear equations form the backbone of algebra. They are equations that represent straight lines when graphed on a coordinate plane. In these equations, variables are raised only to the power of one and involve coefficients and constants. Here are some characteristics:

• The general form of a linear equation in two variables is \( ax + by = c \).
• The solutions to these equations are pairs of numbers that satisfy both equations simultaneously.
• In the exercise, both equations \( 3x - 2y = 12 \) and \( 5x = 4 - 2y \) were linear equations.

Understanding the structure of linear equations is crucial, as it allows you to apply methods like substitution effectively. The goal is to simplify until you can identify the point where the lines, or their solutions, intersect.

Equation Simplification

Simplifying equations is key to solving them effectively and accurately. It involves using algebraic manipulation to transform complex equations into simpler forms by:

• Distributing and combining like terms.
• Reducing fractions or using multiplication to eliminate denominators.
• Reorganizing terms to isolate variables.

In the original solution, simplifying \( 3\left(\frac{4 - 2y}{5}\right) - 2y = 12 \) involved distributing the multiplication and combining terms, resulting in \( 12 - 16y = 60 \) after eliminating fractions. This step-by-step simplification was crucial to solve for \( y \), allowing further substitution to find \( x \).
Happy problem solving! Simplification is your friend.

Verification of Solutions

After solving a system of equations, it's vital to verify the solutions to ensure they are correct. Verification involves substituting the found values back into the original equations and checking if they satisfy both equations. This step confirms the accuracy of your solution process. Here's how to verify successfully:

• Substitute the values of the variables back into each original equation.
• Simplify and check if the equation holds true for both.
• If both equations check out, your solution is correct!

In the exercise, substituting \( x = 2 \) and \( y = -3 \) back into the original equations showed that both equations balanced. This process proved that our solutions were indeed correct. Remember, solution verification protects against calculation errors and confirms the validity of results.