Problem 24 - Introductory Algebra for College Students

Question

In Exercises \(1-44,\) solve each system by the addition method. If there is no solution or an infinite number of solutions, so state. Use set notation to express solution sets. $$\left\\{\begin{array}{l} 5 x=4 y-8 \\ 3 x+7 y=14 \end{array}\right.$$

Step-by-Step Solution

Answer

The solution to the system of equations is \(x= -0.23429\) and \(y= 1.31429\)

Stepping Up The Equations

Firstly, step up both equations in a way that the variables align vertically. The given equations are:\(5x = 4y - 8\) and \(3x + 7y = 14\)Rewrite the first equation as \(5x + 0y = 8\) so that y terms align vertically.

Step 2: Conversion

The aim now is to remove one variable by adding the two equations. To do this, convert one variable's coefficients into the other variable's. Here, multiply the first equation by 3 and the second equation by 5. \(15x + 0y = 24\)\( 15x + 35y = 70\)

Step 3: Subtraction

To get rid of the x variable, subtract the first equation from the second:\(15x + 35y - (15x + 0y) = 70 - 24\)This will result in:\(35y = 46\)

Step 4: Solving For y

To solve for y, divide both sides of the equation by the coefficient of y, 35:\(y = 46/35\)Simplify the fraction to get \(y = 1.31429\)

Step 5: Solving For x

Substitute y = 1.31429 into the first equation given:\(5x= 4(1.31429) - 8\)Solving this equation yields \(x= -0.23429\)

Key Concepts

System of Equations

When faced with the challenge of solving a system of equations, it's crucial to understand that you're dealing with two or more equations simultaneously. Each equation represents a line, and the solution to the system is the point or points where these lines intersect. In our exercise, we have:

• Equation 1: \(5x = 4y - 8\)
• Equation 2: \(3x + 7y = 14\)

The goal is to find values for \(x\) and \(y\) that satisfy both these equations at the same time. Letting one equation be dependent on the variables \(x\) and \(y\), while solving for the other, helps in identifying the solutions more clearly.
By employing the addition method, we manipulate these equations to eliminate one variable, making it easier to solve for the other. It's a systematic approach that relies heavily on aligning and combining the equations smartly.

Variables Alignment

Aligning variables is a crucial initial step in solving systems of equations using the addition method. This method requires that variables are aligned vertically, so that they can be easily added or subtracted.
In our exercise, we initially have:\(5x = 4y - 8\). By rewriting this equation as \(5x + 0y = 8\), the variable \(y\) is aligned with that in the second equation, \(3x + 7y = 14\).
This alignment simplifies the process of eliminating variables by ensuring that like terms are directly above or below each other. It's like setting the table before a dinner---getting everything in place ensures the rest of your work is streamlined and effective.

Solving Linear Equations

Solving linear equations involves finding the values of the variables that make the equation true. Once our variables are aligned, we can proceed with eliminating one variable by making their coefficients equal.

• First, we multiplied the first equation by 3 and the second by 5:\[15x + 0y = 24\]\[15x + 35y = 70\]
• This transformation creates equivalent equations that retain the same solution set.
• We then subtract the first equation from the second to eliminate the \(x\) variable:\[35y = 46\]

By isolating \(y\), we simplified the system to one variable, which we then solved by dividing both sides by 35, finding \(y = 1.31429\). The final step is solving for \(x\) using the value of \(y\) in one of the original equations. By substituting \(y\) back into \(5x = 4y - 8\), we solved for \(x\) and discovered it equals \(-0.23429\).

Set Notation

Once you have the values for \(x\) and \(y\), it's useful to express the solution in set notation. Set notation is a standardized mathematical language that helps convey solutions neatly and clearly. In this exercise, we found the solutions: \(x = -0.23429\) and \(y = 1.31429\).
Using set notation, the solution to the system of equations is expressed as:

• \( \{(x, y) | x = -0.23429, y = 1.31429\} \)

This notation reads as "the set of \(x\) and \(y\) such that \(x = -0.23429\) and \(y = 1.31429\)." This not only communicates that \(x\) and \(y\) are specific values but also clearly states that these values are the only solutions to the system. It can often help avoid confusion by clearly delineating what the solution includes and does not include.