Solving a System of Linear Equations: A Step-by-Step Guide with Examples

Solving a system of linear equations is a fundamental skill in algebra. This article will guide you through the process of solving a system of two linear equations using the substitution method. We'll also provide detailed examples to help you understand the concept better.

Introduction to Linear Equations

A linear equation is an equation that forms a straight line when plotted on a graph. It can be expressed in the form ax by c, where a, b, and c are constants, and x and y are variables. When dealing with two variables, we often have two linear equations.

Setting Up the Equations

Let's consider the following system of equations:

x y 12 3x 5y 44

The first step is to understand what these equations represent. In this case, the first equation states that the sum of the two numbers is 12, and the second equation states that three times the first number plus five times the second number equals 44.

Step-by-Step Solution Using the Substitution Method

We will use the substitution method to solve this system of equations. This involves solving one of the equations for one variable and then substituting that expression into the other equation. Here’s how it works:

Step 1: Solve One Equation for One Variable

Let's start by solving the first equation for y:

x y 12 Subtract x from both sides: y 12 - x

This gives us the expression for y in terms of x.

Step 2: Substitute the Expression into the Second Equation

Now, substitute y 12 - x into the second equation:

3x 5y 44 3x 5(12 - x) 44 Distribute the 5: 3x 60 - 5x 44 Combine like terms: -2x 60 44 Subtract 60 from both sides: -2x -16 Divide by -2: x 8

Step 3: Solve for the Second Variable

Now that we have x 8, we substitute it back into the expression we found for y from Step 1:

y 12 - x y 12 - 8 Simplify: y 4

Conclusion

The two numbers are x 8 and y 4. Let's verify our solution:

Sum: 8 4 12 Equation Check: 3(8) 5(4) 24 20 44

Both conditions are satisfied, confirming our solution.

Additional Examples

Let's go through a few more examples to solidify your understanding:

Example 1

Given:

ab 12 3a 5b 44

We can manipulate the first equation as follows:

ab 12 Multiply the first equation by 3 and the second by 1: 3a 3b 36 3a 5b 44 Subtract the first equation from the second: (3a 5b) - (3a 3b) 44 - 36 2b 8 b 4 Substitute b 4 into the first equation: a4 12 a 3

Thus, the numbers are a 3 and b 4.

Example 2

Given:

ab 12 3a 5b 44

We can use similar steps:

ab 12 Multiply the first equation by 3 and the second by 1: 3a 3b 36 3a 5b 44 Subtract the first equation from the second: 2b 8 b 4 Substitute b 4 into the first equation: a4 12 a 3

Therefore, the numbers are a 3 and b 4.

Summary

Solving a system of linear equations is a crucial skill in algebra. The substitution method is one of the most effective techniques for solving such systems. By following the steps outlined in this article, you can solve similar problems accurately and efficiently. Practice regularly to reinforce your understanding and improve your skills.