Problem 28 - Algebra 1: Concepts and Skills

Question

Use the substitution method or linear combinations to solve the linear system and tell how many solutions the system has. Then describe the graph of the system. $$\begin{array}{l} {-6 x+2 y=-2} \\ {-4 x-y=8} \end{array}$$

Step-by-Step Solution

Answer

The system of equations has one unique solution that satisfies both equations. The graph of the system would show two intersecting lines at this solution point.

Step 1: Simplify the equations

Simplify the equations by making x the subject of equation in each. The first equation simplifies to \(x = \frac{2y + 2}{6}\), and the second to \(x = \frac{y + 8}{4}\).

Step 2: Use the substitution method

Quite simply, substitution involves replacing one variable in one equation with the equivalent term from the other equation. In this case, set \(\frac{2y + 2}{6}\) equal to \(\frac{y + 8}{4}\) and solve for y.

Step 3: Solve for the other variable

Substitute the value of y into either of the original equations and find the corresponding x value. This provides the corresponding x value, completing the solution to the system of equations.

Step 4: Number of solutions and graph

The number of solutions is determined by how many unique (x, y) pairs satisfy both equations. If there is exactly one pair, the system has one solution, and the graph will show the two lines intersecting at that point. If there are infinitely many pairs, the system has infinitely many solutions, and the graph will show the two lines laying on top of each other. If there are no pairs, the system has no solutions, and the graph will show the two lines being parallel and non-intersecting.

Key Concepts

Substitution Method

The substitution method is a straightforward technique for solving systems of linear equations. You begin by solving one equation for one variable. This expresses the variable in terms of the other variable. Once you have this expression, substitute it into the second equation. This way, you're left with a single equation with one variable, making it much easier to solve.

In our exercise, we started by isolating \(x\) in each equation:

• First equation: \(x = \frac{2y + 2}{6}\)
• Second equation: \(x = \frac{y + 8}{4}\)

Next, set these two expressions equal to each other since both are equal to \(x\). Solving this will give us the value of \(y\). Once \(y\) is found, substitute back into one of the original equations to find \(x\). This technique ensures the solution is valid for both equations simultaneously.

Linear Equations

Linear equations describe a straight line in a coordinate plane. They are typically written in the form \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants. In these equations, both \(x\) and \(y\) are raised only to the first power, which keeps the graph straight.

In solving a system of linear equations, you're essentially finding the point where two lines intersect. This point is the solution of the system. The step-by-step simplification of our given exercise equations relies on this principle.

Understanding how these equations work helps you interpret the resulting answers and graphical representations effectively. Whether it's one solution, no solution, or infinitely many, it all stems from the nature of linear equations.

Graph of a System

The graph of a system of equations represents each equation as a line on a two-dimensional plane. The behavior of these lines—and consequently the nature of their intersection—reveals the number of solutions the system has.

For our system:

• If the lines intersect at exactly one point, the system has one unique solution.
• If the lines are parallel and never meet, there are no solutions.
• If the lines overlap entirely, there are infinitely many solutions.

In our exercise, determining the graphical relationship aids in visualizing the solution. Seeing whether the lines overlap, cross at a single point, or never touch can immediately signal how many solutions exist.