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Algebra Foundations

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

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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.

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.

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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 Foundations |

Type of Tutorial | Animation |

Key Vocabulary | exactly one solution, infinitely many solutions, no solutions |