Problem 6 - Elementary Algebra

Question

Use the elimination-by-addition method to solve each system. $$\left(\begin{array}{l}x-4 y=29 \\ 3 x+2 y=-11\end{array}\right)$$

Step-by-Step Solution

Answer

The solution is \((1, -7)\).

Step 1: Align the Equations

First, take the given system of equations: \( x - 4y = 29 \) and \( 3x + 2y = -11 \). Identify which variable to eliminate; here we choose to eliminate \( y \).

Step 2: Equalize the Coefficients of y

To eliminate \( y \), the coefficients in both equations must be equal and opposite. Multiply the first equation by 1 (it remains \( x - 4y = 29 \)) and the second by 2 to get \( 6x + 4y = -22 \).

Step 3: Add the Equations

Add the modified equations to eliminate \( y \):\[(x - 4y) + (6x + 4y) = 29 + (-22)\] This results in:\[7x = 7\]

Step 4: Solve for x

Divide both sides of the equation \( 7x = 7 \) by 7 to solve for \( x \):\[x = \frac{7}{7} = 1\]

Step 5: Substitute to Find y

Substitute \( x = 1 \) back into one of the original equations to solve for \( y \). Using the first equation:\[1 - 4y = 29\] Rearrange to solve for \( y \):\[-4y = 29 - 1 = 28 \Rightarrow y = -\frac{28}{4} = -7\]

Step 6: Solution Verification

Verify the solution \((x, y) = (1, -7)\) by substituting it into the second equation:\[ 3(1) + 2(-7) = 3 - 14 = -11 \]Since the equation holds true, the solution is verified.

Key Concepts

System of Equations

A system of equations consists of two or more equations that share the same set of variables. These equations are considered simultaneously to find solutions that satisfy all of them. Solving a system of equations means finding the values of the variables that satisfy each equation in the system. This process often involves manipulating the equations using algebraic techniques to simplify or rearrange them.

In the given exercise, the system is defined by two linear equations:

• \( x - 4y = 29 \)
• \( 3x + 2y = -11 \)

Both equations involve the variables \( x \) and \( y \). Our goal is to find the values of \( x \) and \( y \) that make both equations true. Methods like substitution or elimination are commonly used to achieve this.

Solving Linear Equations

Linear equations are equations where the highest power of the variable is one. These equations form straight lines when graphed on a coordinate plane. Solving linear equations involves finding the value of the variable that makes the equation true.

When tackling a system of linear equations, solving for one variable is often the first step. For instance, to solve the equation \( 7x = 7 \), we isolate \( x \) by dividing both sides by 7, resulting in \( x = 1 \).

The solution, \( x = 1 \), is then substituted back into one of the original equations to find the value of the other variable, \( y \). By substituting into the equation \( x - 4y = 29 \), we simplify to find \( y = -7 \). Thus, the solution is \( (x, y) = (1, -7) \). This pair represents the point where both equations intersect on a graph, where both statements are satisfied.

Algebraic Techniques

Algebraic techniques are fundamental in solving systems of equations. One effective technique used is the elimination method, often preferred for its straightforward approach.

In the elimination method, the goal is to eliminate one of the variables by adding or subtracting the equations. This often requires adjusting the coefficients of the variables. In our example, we aimed to eliminate \( y \) by making their coefficients equal and opposite:

• First equation is left as \( x - 4y = 29 \).
• Second equation is multiplied by 2 to become \( 6x + 4y = -22 \).

By adding these equations together, the \( y \) terms cancel out, simplifying the system to a single equation in terms of \( x \).

After solving for \( x \), we backtrack to find \( y \) using substitution. It's crucial to verify the solution is correct in both original equations to ensure no errors were made.