Lesson 7MA 152Section 1.1 and 1.2 (part 1)
An equation is a statement indicating that two quantities are equal. Each unknown (represented by a letter) in an equation is a variable. A solution ( or root) of an equation is a value that makes a true statement when replaced for the variable. A solution set is the set of all solutions. To solve an equation is to find the solution(s).
No value can be a solution of an equation that makes a denominator equal zero. Any equation that has a variable in a denominator may have restrictions on what value or values that may replace it. For example; in the equation , x could not equal 2. The value 2 is restricted from possible values for x.
Some equations are true no matter what value is replaced for the variable. Such an equation is called an identity. For example is an identity. Some other equations, the variable may never be replaced by any value to make the equation true. This type of equation is called a contradiction. An example of a contradiction is the equation . An equation that has a finite number of solutions is a conditional equation. An example of a conditional equation is (solution is -7).
A linear equation is a first-degree polynomial equation and can be written in the form
Ex 1: Solve each equation and categorize.
Ex 2: Solve each equation.
A rational equation is one that contains one or more rational expressions (fractions). Remember: no solutions can make zero denominators.
Ex 3: Solve each.
A formula is an equation with more than one variable (sometimes several variables). Sometimes a formula may be written in as an equivalent in terms of a different variable. For example.
Ex 4: Solve each formula for the given variable.
Now, we will cover the following types of application problems that will be designed with an equation.
- Number problems
- Geometric problems
In the next lesson, we will continue with application problems.
The textbook has a good strategy for solving application problems. I usually recommend the following 5-step strategy.
1.Perhaps draw a picture or make a table after studying the problem.
2.Define the variable (write a sentence that defines what x represents).
2.Find a plan, formula, or sentence that will help you write an equation.
3.Solve the equation.
4.Answer the question asked in the problem.
Solve each problem by writing an equation.
1.Find three consecutive even integers so that the first added to twice the second is the same as twice the third.
2.Marvin took 5 tests (each out of 100 points) during a semester. He made the same score on the first test and the fifth test. Scores on the second, third, and fourth tests were 72, 85, and 79. If Marvin averaged 78, what was Marvin's score on the first test?
3.A caterer charges $45 plus $8 per person. How many persons can Dawn have at her catered dinner, if she has budgeted no more than $140?
4.The area of a triangular swimming pool (right triangle) with a base of 16 feet and height of 20 feet is doubled by adding a rectangular pool that shares a side with the 20 feet side of the right triangle. (See picture) What is the width of the rectangular part of the pool?
5.Find the dimensions of a rectangle with a perimeter of 54 meters, if its length is 3 meters less than twice its width.
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