Problem 48 - Algebra for College Students

Question

Solve each system by using either the substitution method or the elimination- by-addition method, whichever seems more appropriate. \(\left(\begin{array}{l}y=4 x-24 \\ 7 x+y=42\end{array}\right)\)

Step-by-Step Solution

Answer

The solution is \(x = 6\) and \(y = 0\).

Step 1: Identify the Method

The system of equations is given as: \( y = 4x - 24 \) and \( 7x + y = 42 \). Since the first equation is already solved for \(y\), substitution seems to be the straightforward method to use.

Step 2: Substitute for y

Substitute \(y = 4x - 24\) from the first equation into the second equation. This yields the equation: \( 7x + (4x - 24) = 42 \).

Step 3: Combine Like Terms

Combine the \(x\) terms in the equation from Step 2: \( 7x + 4x - 24 = 42 \) simplifies to \( 11x - 24 = 42 \).

Step 4: Solve for x

To isolate \(x\), first add 24 to both sides: \( 11x = 66 \). Next, divide both sides by 11: \( x = 6 \).

Step 5: Substitute Back to Find y

Use the value of \(x\) in the first equation to find \(y\). Substitute \(x = 6\) into \(y = 4x - 24\): \(y = 4(6) - 24 = 24 - 24 = 0\).

Step 6: Verify the Solution

Substitute \(x = 6\) and \(y = 0\) into the second equation to check: \(7(6) + 0 = 42\). Since this holds true, the solution is verified.

Key Concepts

Substitution Method

The substitution method is a powerful approach for solving systems of linear equations. It is especially useful when one of the equations is already solved for one variable, making it easier to express this variable in terms of the other. Here’s how the process works:

• Identify which equation is more convenient to solve for a variable, if it is not already.
• Substitute the expression of this variable into the other equation.
• Solve the resulting equation to find the value of one variable.

Once we have the value of one variable, we use it to find the value of the other variable by plugging it back into one of the original equations. This way, we break down a system of equations into more manageable one-variable equations, solving them step-by-step.

Elimination Method

The elimination method, also known as the addition method, involves manipulating the equations in a system to eliminate one of the variables. By adding or subtracting the equations from one another, we can cancel out a variable, which simplifies the system drastically. Here's the core idea:

• Adjust the equations as necessary so that adding or subtracting them will cancel one variable out.
• Solve the resulting single-variable equation.
• Use this solution value to find the other variable.

The elimination method works particularly well when both equations are in standard form, and it can be necessary to multiply one or both equations to align the coefficients for effective elimination.

Linear Equations

Linear equations form the backbone of algebra and explicitly map a straight line when plotted on a graph. Understanding their basic structure is crucial:

• A linear equation in two variables can be expressed in the form: \( ax + by = c \).
• These equations present a constant rate of change, represented by their slope in a graph.
• Solutions to systems of these equations are points where their graphs intersect.

Solving systems of linear equations involves finding these intersections, representing consistent values for the involved variables. Such systems can either yield one solution (intersecting lines), no solution (parallel lines), or infinitely many solutions (the same line represented in different forms).

Algebraic Solution Verification

After solving a system of equations, it’s essential to verify the solution. This ensures that the solved values satisfy the original equations perfectly. This can be done in the following way:

• Substitute the found values back into each of the original equations.
• Check if these substitutions yield true identities, confirming the solution satisfies both equations.

If the solution does not satisfy the equations, re-evaluate the solving steps as errors might have occurred. Verification acts as a crucial step, providing confidence in the correctness of your solution. It is akin to "double-checking" your work, which is always a good practice in math.