Solving Linear Equations using the Substitution Method

Study Notes

Solving Linear Equations using the Substitution Method

The substitution method is a technique used to solve systems of linear equations algebraically. In this approach, you use one equation to find the value of one variable, and then you substitute that value back into the other equation to solve for the remaining variable. This method is particularly useful when dealing with complex equations where finding the intersection points directly through graphing may not be feasible.

Let's consider an example: finding the intersection of two lines represented by the equations x + 2y = 9 and 3x + 5y = 20. Instead of trying to find the intersection point by graphing the lines, we can use the substitution method.

Substitution Method Steps

Solve one of the equations for any variable: In this case, let's solve the first equation for x. Subtracting 2y from both sides, we get x = 9 - 2y.
Substitute the expression found in step 1 into the other equation: Replace all instances of x in the second equation with the expression derived in step 1. So, 3(9 - 2y) + 5y = 20.
Simplify and solve: Expanding and distributing the terms, we get 27 - 6y + 5y = 20. Rearranging, we have -y = 7. Solving for y, we get y = -7.
Substitute the value of y back into the expression found in step 1: Replace all instances of y in the expression x = 9 - 2y with the value of y we found in step 3. So, x = 9 - 2(-7), which simplifies to x = 9 + 14. Therefore, x = 23.

So, the intersection point of the two lines is (23, -7).

Description

Learn how to solve systems of linear equations algebraically using the substitution method. Understand the steps involved, such as solving for one variable, substituting the value back into the other equation, and simplifying to find the intersection point of the lines represented by the equations. Practice solving linear equations with this technique!