Problem 28 - Elementary Algebra

Question

Solve each system by using either the substitution or the elimination-by- addition method, whichever seems more appropriate. $$\left(\begin{array}{c}2 x-3 y=4 \\ y=\frac{2}{3} x-\frac{4}{3}\end{array}\right)$$

Step-by-Step Solution

Answer

The system has infinitely many solutions (the lines are coincident).

Step 1: Identification of Method

Given the system of equations: 1. \( 2x - 3y = 4 \) 2. \( y = \frac{2}{3}x - \frac{4}{3} \)We notice that the second equation is already solved for \( y \), making substitution an efficient approach.

Step 2: Substitution Into the First Equation

Substitute the expression for \( y \) from the second equation into the first equation:\[ 2x - 3\left(\frac{2}{3}x - \frac{4}{3}\right) = 4 \]

Step 3: Simplify and Solve for x

Distribute and simplify the equation:\[ 2x - 2x + 4 = 4 \]Further simplification gives:\[ 0x + 4 = 4 \]Since the variables cancel out and the statement is true, this results in infinite solutions implying the lines are coincident.

Key Concepts

Substitution Method

The substitution method is a powerful way to solve systems of equations. In this method, you solve one equation for one variable and then substitute this expression into the other equation. This simplifies the system to a single equation with one variable, making it easier to solve.

• Step 1: Choose one of the equations, preferably one that is already solved for a single variable (like \( y \) in this exercise).
• Step 2: Substitute this expression into the other equation. This eliminates one of the variables, resulting in an equation with only one variable.
• Step 3: Solve this simplified equation for the remaining variable.
• Step 4: Back-substitute this value into the expression from step 1 to find the value of the other variable.

In the original exercise, the equation \( y = \frac{2}{3}x - \frac{4}{3} \) is already solved for \( y \), making substitution particularly efficient. This approach is often favored when an equation is readily expressed in terms of one variable.

Elimination Method

The elimination method, also known as the addition method, is another strategy for solving systems of equations. This method involves adding or subtracting equations in order to eliminate one of the variables, making it easier to solve for the remaining variable.

• Step 1: Multiply one or both equations by a constant to align the coefficients of one of the variables.
• Step 2: Add or subtract the equations to eliminate one variable.
• Step 3: Solve the resulting equation for the remaining variable.
• Step 4: Substitute the value found back into one of the original equations to solve for the other variable.

While in our exercise, the substitution method was more appropriate, understanding the elimination method is crucial, as it can be more suitable when the equations are already aligned for easy addition or subtraction.

Coincident Lines

Coincident lines occur when a system of linear equations represents the same line. This means they have infinitely many solutions, as they overlap perfectly at every point. Understanding coincident lines is essential for interpreting solutions to systems of equations.

• When using either the substitution or elimination method, if all variables cancel out and you end up with a true statement (like \( 4 = 4 \)), the lines are coincident.
• This implies all points on the one line are also points on the other line.
• This results in infinite solutions, as opposed to a single solution or no solution.

In the exercise, when variables canceled out and \( 0x + 4 = 4 \) was simplified to a true statement, it confirmed the two equations represent coincident lines, indicating they share all points in common.