You currently have JavaScript disabled on this browser/device. JavaScript must be enabled in order for this website to function properly.

ZingPath: Solving Linear Equations

Solution Sets of Linear Equations                              Searching for

Solving Linear Equations

Learn in a way your textbook can't show you.
Explore the full path to learning Solving Linear Equations

Lesson Focus

Solution Sets of Linear Equations

Algebra-1

You get to learn how to find exact solutions to linear equations by using algebraic methods, and approximate solutions by using graphical methods.

Now You Know

After completing this tutorial, you will be able to complete the following:

• Know that linear equations of the form ax + b = cx + d can have exactly one solution, no solutions, or infinitely many solutions.

Everything You'll Have Covered

Consider the linear equation ax + b = cx + d. How many possible solutions are there for such an equation?

~ There are exactly three possibilities. If a is not equal to c, then the equation has exactly one solution. If a = c and b does not equal d, then there is no solution. If a = c and b = d, then there are infinitely many solutions.

How many solutions are there to the linear equation 3x + 2 = 3x + 6?

~ Considering this equation as an equations of the form of ax + b = cx + d, we have that a = 3, b = 2, c = 3, and d = 6. Since a = c and b does not equal d, the equation has no solution.

How many solutions are there to the linear equation -3x + 2 = -3x + 2?

~ Considering this equation as an equaion of the form ax + b = cx + d, we have that a = -3, b = 2, c = -3, and d = 2. Since a = c and b = d, the equation has infinitely many solutions.

How many solutions are there to the linear equation 3x + 2 = 2x + 6?

~ Considering this equation as an equation of the form ax + b = cx + d, we have that a = 3, b = 2, c = 2, and d = 6. Since a does not equal c, the equation has exactly one solution.

Choose one of the possibilities from question 1 and explain why this occurs under the required restrictions on a, b, c, and d.

~ Answers will vary. In each case, we can use inverse operations to illustrate. When a does not equal c, we use inverse operations to solve the linear equation: We see that this is the only value of x that satisfies the equation, so the equation has only one solution. When a = c and b does not equal d, we can rewrite the equation ax + b = cx + d as ax + b = ax + d. Now we subtract ax from both sides. We see that: This is a false statement since b does not equal d. Therefore, there can be no value of x such that ax + b = cx + d when a = c and b does not equal d, so the equation has no solution. Finally, when a = c and b = d, we can rewrite the equation ax + b = cx + d as ax + b = ax + b. Now we subtract ax from both sides. We see that: This is always true regardless of the choice of x. Therefore, every value of x makes the equation ax + b = cx + d true when a = c and b = d, so the equation has infinitely many solutions. "

Tutorial Details

 Approximate Time 2 Minutes Pre-requisite Concepts Students should have an intellectual grasp on the terms: exactly one solution, infinitely many solutions, and no solutions. Course Algebra-1 Type of Tutorial Animation Key Vocabulary exactly one solution, infinitely many solutions, no solutions