Problem 14 - Intermediate Algebra

Question

For Problems \(1-18\), use the elimination-by-addition method to solve each system. (Objective 1 ) $$ \left(\begin{array}{lr} 2 x-7 y= & -2 \\ 3 x+y= & 1 \end{array}\right) $$

Step-by-Step Solution

Answer

The solution is \( x = \frac{5}{23} \), \( y = \frac{8}{23} \).

Step 1: Align the Equations

First, write down the given system of equations with like terms aligned. The given system is: \[ \begin{align*} 2x - 7y &= -2 \ 3x + y &= 1 \end{align*} \] We can easily proceed with elimination since the equations are aligned.

Step 2: Eliminate One Variable by Addition

To eliminate one of the variables, multiply the second equation by 7 to make the coefficients of \( y \) equal in magnitude but opposite in sign. The system becomes: \[ \begin{align*} 2x - 7y &= -2 \ 21x + 7y &= 7 \end{align*} \] Now, add the two equations together to eliminate \( y \): \[ \begin{align*} (2x - 7y) + (21x + 7y) = -2 + 7 \ 23x = 5 \end{align*} \]

Step 3: Solve for the Remaining Variable

Now solve the resulting equation for \( x \): \[ x = \frac{5}{23} \] This gives the value of \( x \).

Step 4: Substitute Back to Find the Other Variable

Substitute \( x = \frac{5}{23} \) back into one of the original equations to find \( y \). Using the second equation, \( 3x + y = 1 \): \[ 3\left(\frac{5}{23}\right) + y = 1 \] \[ \frac{15}{23} + y = 1 \] \[ y = 1 - \frac{15}{23} \] \[ y = \frac{23}{23} - \frac{15}{23} \] \[ y = \frac{8}{23} \]

Step 5: Verify the Solution

Substitute \( x = \frac{5}{23} \) and \( y = \frac{8}{23} \) into the first original equation to verify correctness: \[ 2\left(\frac{5}{23}\right) - 7\left(\frac{8}{23}\right) = -2 \] \[ \frac{10}{23} - \frac{56}{23} = -\frac{46}{23} = -2 \] This confirms the solution is correct.

Key Concepts

Elimination Method

The elimination method is a powerful technique used for solving systems of linear equations. It involves adding or subtracting equations to eliminate one of the variables, making it easier to solve for the remaining variable(s).
Here’s how it works:

• First, we align the equations vertically by their terms, ensuring that the variables and constants are properly positioned.
• Next, we modify the equations, if necessary, to create opposite coefficients for one of the variables. This allows us to add or subtract the equations and eliminate that variable.
• We then perform the arithmetic operation (addition or subtraction) to remove the chosen variable.
• With one variable eliminated, we solve the resulting simpler equation for the remaining variable.
• Finally, we use the found value to substitute back into one of the original equations to find the value of the eliminated variable.

The elimination method is particularly useful when dealing with equations that are easily manipulated to allow one variable to cancel out.

System of Equations

A system of equations consists of multiple equations that share some variables. The goal is to find one solution that satisfies all the equations in the system simultaneously.
Solving a system can be done using various methods, like graphing, substitution, and elimination.

• Graphing method: Involves plotting each equation on a graph to find their point of intersection, which represents the solution.
• Substitution method: Involves solving one of the equations for a single variable and substituting this into the other equations.
• Elimination method: Utilizes addition or subtraction to remove a variable, simplifying the system effectively.

The system can be consistent (with at least one solution), inconsistent (no solutions), or dependent (infinite solutions). Understanding the nature of the system helps us choose the most effective solving method.

Solving Linear Equations

Solving linear equations involves finding the value of the variables that make the equation true. A linear equation typically forms a straight line when graphed and has no exponents higher than one.
The key steps in solving linear equations include:

• Combining like terms to simplify the equation, if needed.
• Using inverse operations to isolate the variable. For instance, if a variable is added to a number, we subtract to cancel it; if it is multiplied, we divide, and so on.
• Each operation must be done on both sides of the equation to maintain balance.
• Conducting a quick check by substituting the solution back into the original equation to ensure correctness.

These steps form the foundation of solving not only simple equations but also more complex systems involving multiple variables.

Substitution Method

The substitution method is another technique for solving systems of equations, often used when one equation is easily solved for one variable.
Here’s a step-by-step process for using the substitution method:

• First, solve one of the equations for one of the variables in terms of the other variables. This means rearranging one equation to express a variable like \( x = ... \).
• Next, substitute this expression into the other equation(s). This replaces the variable with the expression and results in an equation with only one variable.
• Solve this new equation to find the value of the single variable.
• Finally, substitute this value back into the expression found in the first step to solve for the other variable.

This method is particularly useful when one equation is already isolated or can be easily manipulated to isolate a variable. The substitution method provides a straightforward process for solving systems, especially when equations are simple enough to handle algebraically.