Problem 68 - Algebra 1

Question

Solve the system of linear equations if possible. Does the system have exactly one solution, no solution, or infinitely many solutions? $$ \begin{aligned}&8 x+4 y=-4\\\&4 x-y=-20\end{aligned} $$

Step-by-Step Solution

Answer

The solution to the system of equations is \(x = -\frac{7}{2}\) and \(y = \frac{15}{4}\). Thus, the system has exactly one solution.

Step 1: Synchronize Coefficients

First, to solve the system of linear equations, in order to make the x-coefficients in both equal, we multiply the second equation by 4, which gives us: \(16x–4y=-80\). Now, our equations look like this: \[ \begin{aligned} &8x+4y=-4 \ &16x-4y=-80 \end{aligned}\]

Step 2: Sum Equations and Solve for x

We then add the two equations together. This gives us: \(24x=-84\). Solving this equation for x, we divide both sides by 24, which gives \(x=−\frac{7}{2}\)

Step 3: Substitute x into First Equation and Solve for y

Substitute \(x=−\frac{7}{2}\) into the first equation \(8x+4y = -4\), we have \(8(-\frac{7}{2})+4y = -4\). Solving this for y we get \(y = \frac{15}{4}\)

Key Concepts

Linear Equations

Linear equations are mathematical statements that show the equality between two expressions composed of variables and constants. They are called 'linear' because their graphed representations are straight lines on a coordinate plane. Each linear equation represents a relationship where every variable is raised only to the first power.

When dealing with a system of linear equations, like in our exercise, we are looking at multiple linear equations that we solve simultaneously to find common solutions for the variables involved. Systems of linear equations can be categorized based on the number of solutions they possess:

• A unique solution, where lines intersect at one point.
• No solution, when the lines are parallel and never intersect.
• Infinitely many solutions, where the lines lie on top of each other, meaning all points are solutions.

Our exercise presents a two-equation system to be solved, where the goal is to find the specific values for the variables x and y that satisfy both equations at the same time.

Substitution Method

The substitution method is one of the techniques for solving systems of linear equations. This method involves solving one of the equations for one variable and then substituting that solution into the other equation. It's especially useful when one of the equations is easy to solve for one of the variables.

Let's take a closer look at how the substitution method could be used in our example. If we solve the second equation, 4x - y = -20, for y, we get y = 4x + 20. We could then substitute this expression for y into the first equation, 8x + 4y = -4, which would then only contain the variable x. Once we solve this new equation for x, that value could be substituted back into y = 4x + 20 to find the value of y. Our step-by-step solution, however, utilizes the elimination method, which is often more efficient when equations are already in a form that facilitates elimination.

Elimination Method

In contrast to the substitution method, the elimination method focuses on removing one variable to make it possible to solve for the other. To achieve this, we manipulate the equations such that adding or subtracting them would result in one of the variables being canceled out.

In the step-by-step solution provided, we multiplied the second equation by 4, changing its form but not its solution set, in order to synchronize the coefficients of one variable. The resultant equations thus allowed us to eliminate the variable y by adding both equations together. Having found the value for x, we didn't need to use substitution in this specific case. Instead, we just needed to replace x with its value in either of the original equations to solve for y. This is the beauty of the elimination method: once the coefficients are aligned, these steps can quickly lead to the solution of both variables.