Equations
An equation is a statement that connects two expressions with the equals sign = to express their equality.
Solving Equation by Using Different Operations
Equations may be satisfied for some values, but they may be unsatisfied for others. A solution is a value that “satisfies” an equation. The equation equality can be proved by using operations like addition, subtraction, multiplication and division.
Properties
Addition Property of Equality
The two sides of an equation remain equal when the same number is added to each side.
If a = b then a + c = b + c
Subtraction Property of Equality
The two sides of an equation remain equal when the same number is subtracted to each side.
If a = b then a – c = b – c
Multiplication Property of Equality
When both sides of an equation are multiplied by the same nonzero number, the two sides remain equal.
If a = b then a . c = b . c
Multiplicative Inverse Property
A non-zero number a and its reciprocal, \( \frac{1}{a} \), have a product of one.
\( \Rightarrow n.\frac{1}{n}=\frac{1}{n}.n=1,~n\neq 0 \)
Numbers
\( \Rightarrow 8.\frac{1}{8}=1 \)
Division Property of Equality
When both sides of an equation are divided by the same nonzero number, the two sides remain equal.
If a = b then \( \frac{a}{c}=\frac{b}{c},~c\neq 0 \)
Solved Examples
Example 1: Solve the following two – step equations using different operations.
(a) – 4a + 6 = 2
(b) \( \frac{x}{4}-\frac{1}{4}=\frac{9}{2} \)
Solution:
(a) We have: – 4a + 6 = 2
– 4a + 6 = 2 Using Subtraction Property of Equality
– 6 – 6
– 4a = -4
– 4a = -4 Using Division Property of Equality
\( \div \) 4 \( \div \)4
a = 1
The solution is a = 1.
(b) We have \( \frac{x}{4}-\frac{1}{4}=\frac{9}{2} \)
\( \frac{x}{4}-\frac{1}{4}=\frac{9}{2} \) Using addition property of equality
\( +\frac{1}{4} \) \( +\frac{1}{4} \)
\( \frac{x}{4}=\frac{19}{4} \)
\( \frac{x}{4}\times 4=\frac{19}{4}\times 4 \) Using multiplication property of equality
x = 19
The solution is, x = 19.
Example 2: Solve the following two – step equations by combining like terms.
(a) – 3a + 6a = 21
(b) \( \frac{x}{2}-\frac{x}{4}=9 \)
Solution:
(a) We have: – 3a + 6a = 21
3a = 21 Combining the given like terms
a = 7 Divide both sides by 3
The solution is a = 7.
(b) We have: \( \frac{x}{2}-\frac{x}{4}=9 \)
\( \frac{x}{4}=9 \) Combining the given like terms
x = 36 Multiply both sides by 4
The solution is x = 36.
Example 3: The perimeter of the rectangular swimming pool is 800 ft. The pool is 100 feet long and 100 feet wide. What is the length of the pool’s measurement?
Solution:
You’ve been given a rectangular swimming pool with a perimeter of 800 feet and a width of 100 feet. You’ve also been given the task of determining the length of the pool.
Make a sketch of the swimming pool. Then, using the formula for the perimeter of a rectangle, write and solve the equation to find the length of the pool.
Perimeter of the rectangle (P) is 2l + 2w
Substitute the value of P and w
800 = 2l + 2 ( 100 )
800 = 2l + 200
2l = 600 Subtract 200 from both sides
l = 300 Divide by 2
Hence, the length of the swimming pool is 300 feet.
Use the subtraction and addition properties to solve equations:
Step 1: Replace the variable in the equation with the number.
Step 2: Simplify both sides of the expressions of an equation.
Step 3: Check to see if the resulting equation is correct.
A number is a solution if it is true.
Using one variable to solve linear equations:
Step 1: Using LCM, remove any fractions that may exist.
Step 2: Make both sides of the equation as simplified as possible.
Step 3: Identify and isolate the variable.
Step 4: Double-check your answer
When solving multi-variable, multi-step equations, the first rule is to make sure you have the same number of equations as the number of variables in the equations.
Then, for one of the variables, solve one of the equations and plug that expression into the other equation for what it equals.
A multivariable function is simply a function with multiple numbers as an input or as an output. A single-variable function, on the other hand, has a single-number input and single-number output.