Solving Linear Equations
Property of Equality
2. Solving Linear Equations
3. Solving for One Variable in a Formula
4. Summary: Process for Solving Linear Equations
5. Worked out Solutions for Exercises
An equation is a statement that two expressions have the same value:
Any number or the value of the variable that makes an equation a true statement is called a solution of the equation. The process of finding the solution of an equation is called solving the equation for the variable or unknown. You can also call it finding x (or y, or whatever variable you are using). All solutions should be checked by substituting back into the original equation, and seeing if that will give a true statement. If you solved [pic] to give you [pic], you will easily see if you have the right answer by substituting [pic] back into [pic] (you don’t).
Linear equation in one variable (with one unknown):
A linear equation in one variable can be written in the form
[pic], where a, b, and c are real numbers and a[pic]0.
Note that the variable (x) is raised to the first power: that’s how you recognize a linear equation.
Examples of linear equations:[pic]
[pic] can be manipulated to be written in the standard form of [pic]
Property of Equality:
If a, b, and c are real numbers, [pic] are equivalent equations
[pic] are equivalent equations
[pic] are equivalent equations, [pic]
This property guarantees that by adding, subtracting (as in adding a negative), multiplying or dividing by the same quantity on both sides you end up with equivalent equations and the solution of the equation is not changed.
Just remember: Whatever you do to one side, you must do to the other side!
4.2 Solving Linear Equations
Example 1:Solve [pic] for x.
Solution: To solve for x, we want x alone on one side of the equation. To do this, we add 7 to both sides of the equation.
To check, substitute 17 in for x in the original equation:[pic]
Since the check gives a true statement, the solution is 17. You will see the solution written in several ways:x=17,17, [pic], the answer is 17. S is the set of all solutions, and it contains only the element 17. In a formal class, it usually depends on the book used which form of the answer you chose. For testing purposes, just be aware that may you see any of the above ways.
Example 2:Solve [pic] for x.
Solution:To solve for x, we want x alone on one side of the equation. [pic]means [pic], so we divide both sides by the coefficient of x:
Conclusion:The answer is [pic].
Exercise 1:Solve [pic] for x and check your solution.
Exercise 2:Solve [pic] for x and check your solution.
Example 3:Solve[pic] for y.
Conclusion:The answer is [pic]
Exercise 3:Solve [pic] for x and check your solution.
Example 4:Solve[pic] for t.
Conclusion:The answer is –7.
How would you simplify [pic]?
Most people would simply use the order of operations: first add what is inside the parenthesis (2 plus 1 is 3), and then multiplying the result by 4 (which gives 12). However, you could multiply the 2 by 4 (that’s 8), the 1 by 4 (that’s 4), and then add the results together (which gives 12). This last method is called using the distributive property: it “eliminates” parenthesis: