Problem 31 - Elementary Algebra

Question

Solve each system by using either the substitution or the elimination-by- addition method, whichever seems more appropriate. $$\left(\begin{array}{c}y=\frac{2}{3} x-4 \\ 5 x-3 y=9\end{array}\right)$$

Step-by-Step Solution

Answer

x = -1, y = -\frac{14}{3}

Step 1: Identify Method of Solution

Examine the system of equations: \( y = \frac{2}{3}x - 4 \) and \( 5x - 3y = 9 \). Here, the substitution method seems more suitable as \( y \) is already isolated in the first equation, which makes substituting in the second equation straightforward.

Step 2: Substitute Expression for y

Take the expression for \( y \) from the first equation, \( y = \frac{2}{3}x - 4 \), and substitute it into the second equation \( 5x - 3y = 9 \). This gives us:\[ 5x - 3\left(\frac{2}{3}x - 4\right) = 9 \]

Step 3: Simplify the Equation

Distribute the -3 in the equation:\[ 5x - 2x + 12 = 9 \]Combine like terms to simplify:\[ 3x + 12 = 9 \]

Step 4: Solve for x

Subtract 12 from both sides of the equation to isolate the term involving \( x \):\[ 3x = 9 - 12 \]\[ 3x = -3 \]Divide each side by 3 to solve for \( x \):\[ x = -1 \]

Step 5: Substitute x Value Back into y Equation

Use the value \( x = -1 \) in the first equation to find \( y \):\[ y = \frac{2}{3}(-1) - 4 \]Calculate:\[ y = -\frac{2}{3} - 4 \]\[ y = -\frac{2}{3} - \frac{12}{3} \]\[ y = -\frac{14}{3} \]

Step 6: State the Solution

The solution to the system of equations is \( x = -1 \) and \( y = -\frac{14}{3} \).

Key Concepts

Substitution Method

The substitution method is a useful approach for solving systems of equations, especially when one of the equations has already isolated a variable. In the given exercise, the first equation is expressed as \( y = \frac{2}{3}x - 4 \). This makes it convenient to substitute \( y \) in the second equation.
Substitution works by replacing one variable with its equivalent expression from another equation. This single equation can then be solved for one variable. Once you find the value of this variable, you can substitute it back into one of the original equations to find the other variable. Here are the steps involved in the substitution method:

• Identify which equation makes substitution easier (usually, this is when one variable is isolated).
• Substitute the expression into the other equation.
• Solve the resulting equation.
• Substitute back to find the second variable.

Let's see these steps applied: We took the expression \( y = \frac{2}{3}x - 4 \) and substituted it into the second equation, eliminating \( y \) entirely and allowing us to focus on solving for \( x \). Once \( x \) was found, we substituted back to find \( y \). This method is helpful when substitution immediately simplifies one of the equations.

Elimination by Addition

Elimination by addition, also known as the elimination method, is another strategy for solving systems of equations. It involves adding or subtracting equations to eliminate one of the variables, which makes solving for the other variable more straightforward.
In cases where the coefficients don't allow direct cancellation, you can multiply one or both equations by a number to align coefficients. Here are the general steps:

• Align the equations so that the coefficients of one variable match (or are negatives of each other).
• Add or subtract the equations to eliminate one variable.
• Solve for the remaining variable.
• Back-substitute to find the eliminated variable.

For the given set of equations, although the substitution method was simpler, elimination could have been used by modifying the equations to cancel out \( y \) or \( x \), demonstrating the flexibility of equation-solving techniques.

Solving Equations

Solving equations is a fundamental skill in algebra, where the goal is to find the unknown variables that satisfy given equations. This process involves manipulating equations using algebraic operations until the solution is revealed.
For the given problem, solving began once the substitution was made in:

• Start with simplifying the expression: \( 5x - 3(\frac{2}{3}x - 4) = 9 \).
• Distribute and combine like terms to get a simpler equation: \( 3x + 12 = 9 \).
• Isolate \( x \) by performing the necessary arithmetic operations.
• Finally, solve for \( x \) and substitute back to find \( y \).

Each step requires careful attention to ensure operations are correctly applied, making sure the solution is consistent with the original equations. This involves balancing equations, verifying each step, and sometimes checking the solution by substituting both values back into the original equations.

Linear Equations

A linear equation is an equation that graphs a straight line when plotted on a coordinate plane. These equations have no exponents higher than one, making them relatively simple and straightforward to solve.
In the system provided, both equations are linear:

• \( y = \frac{2}{3}x - 4 \)
• \( 5x - 3y = 9 \)

Each represents a line, and the solution to the system is the point where the two lines intersect. Understanding linear equations allows you to visualize the problem and anticipate the solution method's impact on the system. Such equations are foundational in algebra and applicable across various fields due to their simplicity and ease of interpretation. Recognizing linear equations also helps in predicting whether methods like substitution or elimination are practical.