Problem 8 - Precalculus with Limits

Question

Solving a System by Substitution In Exercises \(7-14,\) solve the system by the method of substitution. Check your solution(s) graphically. $$\left\\{\begin{array}{rr}{x-4 y} & {=-11} \\ {x+3 y} & {=\quad 3}\end{array}\right.$$

Step-by-Step Solution

Answer

The solution to the system of equations is \(x = -3, y = 2.\)

Step 1: Isolate a variable

We start by isolating a variable in one of the equations. For this particular system, it's easiest to isolate \(x\) in either of the two equations. Isolating \(x\) in the first equation gives us \(x = 4y - 11.\)

Step 2: Substitute into the second equation

Now we substitute this into the second equation: \(x + 3y = 3\). Substituting in for \(x\) from above gives us: \(4y - 11 + 3y = 3\). Simplifying, we get \(7y - 11 = 3\). Solving for \(y\), we get \(y = 2.\)

Step 3: Substitute back

Then we can substitute \(y = 2.\) back into the equation from the first step, \(x = 4y - 11\), which gives us \(x = 4*2 - 11 = -3.\)

Key Concepts

Substitution Method

The Substitution Method is a key technique for solving systems of equations. This method involves replacing one variable with an expression derived from another equation. In this exercise, we begin by isolating one of the variables. Isolation means that you solve one equation for one variable in terms of the other variable. Let's isolate \( x \) in the first equation:

• Given: \( x - 4y = -11 \)
• Solving for \( x \): \( x = 4y - 11 \)

Here, we express \( x \) with respect to \( y \) so that \( x = 4y - 11 \). This expression is then substituted into the other equation. Substitution helps reduce the two-variable equations to a single-variable equation. This method can be especially useful if one equation is already solved or easily solvable for one variable.

Algebraic Manipulation

Algebraic Manipulation is a crucial part of solving equations, particularly when using substitution. After substituting the expression for one variable into the other equation, our task becomes simplifying and solving an equation with a single unknown. Let's walk through this:

• The second equation: \( x + 3y = 3 \)
• Substitute for \( x \): \( 4y - 11 + 3y = 3 \)
• Combine like terms: \( 7y - 11 = 3 \)
• Isolate \( y \): \( 7y = 14 \)
• Solve for \( y \): \( y = 2 \)

Manipulating algebraic expressions involves operations such as simplifying, adding, or rearranging terms. This helps to solve for the remaining variable effectively. After finding \( y = 2 \), substitute it back into \( x = 4y - 11 \) to find \( x = -3 \). Each manipulation step is important for correctly finding solutions to the system.

Graphical Solution Verification

Once you solve a system of equations algebraically, verifying the solution graphically is an important step. This ensures that the values found for the variables are correct and consistent. Here's how you can verify a solution graphically:

• Re-write the equations in slope-intercept form \( y = mx + b \).
• For \( x - 4y = -11 \), re-arrange: \( y = \frac{1}{4}x + \frac{11}{4} \)
• For \( x + 3y = 3 \), re-arrange: \( y = -\frac{1}{3}x + 1 \)
• Plot these lines on the graph.

The intersection of the lines on the graph represents the solution of the system: \((x, y) = (-3, 2)\). This graphical check aids in confirming the accuracy of the solution found algebraically, since accurately intersecting lines at the calculated point shows both equations hold true for these values. If they intersect at the right point, you've validated your solution.