Problem 26 - Algebra 1: Concepts and Skills

Question

Choose a solution method to solve the linear system. Explain your choice, and then solve the system. $$ \begin{aligned} &6 x-y=18\\\ &8 x+y=24 \end{aligned} $$

Step-by-Step Solution

Answer

The solution of the system of equations is x = 3 and y = 0.

Step 1: Add the equations

By adding the two equations, 6x - y + 8x + y = 18 + 24. y terms cancel out, which gives us the equation \(14x = 42\)

Step 2: Solve for x

To solve for x, divide the both sides by 14. This gives us the equation \(x = 42 / 14\), and by solving it, we find \(x = 3\)

Step 3: Substitute x in the second equation

Substitute \(x = 3\) into the second equation \(8x + y = 24\). Replace x with 3, this gives us the equation \(8(3) + y = 24\) , which simplifies down to \(24 + y = 24\).

Step 4: Solve for y

To find the value of y, subtract 24 from both sides. This gives us the equation \(y = 24 - 24\), and by solving it, we find \(y = 0\)

Key Concepts

Linear Equations

Linear equations form the backbone of algebra and are among the first types of equations that students encounter. An equation is called linear if it can be expressed in the form of \(ax + by = c\), where \(x\) and \(y\) are variables, and \(a\), \(b\), and \(c\) are constants. The graphs of such equations are always straight lines, hence the term 'linear'.

When solving these equations, we want to find the value of the variables that make the equation true. In the exercise, you were given two such linear equations to solve. The goal was to find the values of \(x\) and \(y\) that satisfy both equations simultaneously. Understanding how to manipulate these equations and use algebraic properties to isolate variables is an essential skill in mathematics.

Systems of Equations

When we have more than one linear equation using the same set of variables, we call it a system of equations. These systems can have one solution, no solutions, or infinitely many solutions. The goal when solving a system is to find the values of the variables that satisfy all equations in the system at the same time.

In our example, you worked with a system of two equations with two unknowns. The addition method, also known as the elimination method, is particularly useful when the coefficients of one variable are opposites, as they were in the given system (\(6x-y\) and \(8x+y\)). By adding the two equations together, the \(y\) terms cancel each other out due to their opposite signs, which simplifies the problem to a single variable equation. This strategy is a clever way to use the structure of the system to your advantage, leading to a quicker solution.

Algebraic Methods

Algebraic methods refer to a range of techniques used to manipulate expressions and solve equations. These methods include addition, subtraction, multiplication, division, factoring, and the use of properties of equality. For systems of linear equations, the chief algebraic methods are substitution, elimination (addition), and graphing.

In the provided exercise, the addition method was chosen because it allowed for an efficient solution. It eliminated one variable, enabling you to solve for the other variable straightforwardly. After finding one variable, you can backtrack to find the other by substituting the known value into either of the original equations. Besides these, the substitution method could be used if one of the equations can easily be solved for one variable and then substituted into the other. Each method has its best-case scenarios, and choosing the most suitable one can greatly simplify the solution process.