Writing Equations with One Variable
A mathematical equation is an statement in which the equal sign, =, is used to show that two expressions are equal to some resultant value.
Expressions
In mathematics, an expression is a sentence that contains at least two numbers and one math operation.
Example:
(1) 2 + 5
(2) a + 6
Equations
A mathematical statement consisting of an equal symbol between two algebraic expressions with the same value is known as an equation.
Examples:
- 2 + 5 = 7
- a + 6 = 7
The image below shows the formation of an equation:
The student has to determine where to place the equal sign in a word sentence written as an equation and look for keywords or phrases such as “is”, “the same as”, or “equals”.
Examples
Example 1: Writing equations using the word sentence.
(a) The sum of the number \(x\) and \(3\) is \(10\)
(b) A number a decreased by \(2\) is \(12\)
(c) \(92\) equals 6 times the number \(x\)
Solution:
Part (a)
We have: The sum of the number \(x\) and \(3\) is \(10\).
“Sum” means addition of numbers.
\(=x+3=10\)
The final equation is x+3=10
Part (b)
We have: The number a decreased by 2 is 12
“Decreased by” means subtraction.
\(=a-2=12\)
The final equation is \(a-2=12\)
Part (c)
We have: 92 equals 6 times a number \(x\)
“Times” means multiplication.
\(92=6x\)
The final equation is \(92=6x\)
Example 2: For a wedding, ten servers will decorate 25 tables.
Each table is set up in the manner depicted. The total number of yellow and purple candles should be c. Which equation can you use to find c?
Solution:
Write an equation using the word sentence of the given question:
Number of tables . Number of candles on each table = Total numbers of candles.
Let candles be represented by c.
\(25.(4+6)=c\)
Example 3: In a spelling bee, 24 students are eliminated after two rounds. Only 96 students are left. Determine the total number of students who participated in the spelling bee.
Solution:
You are given the numbers of students who have been eliminated from a spelling bee and those who have not. You will be asked how many students participated in the spelling bee.
Write and solve an equation establishing a relationship between the number of students who started, the number of students who were eliminated, and the number of students who remained. Use a verbal model as a guide.
Remaining students = Number of students who started playing – Number of students eliminated.
Let the students that started the game in the beginning be denoted as s.
\(96=s-24\)
So, the total number of students (s) = 96 + 24
= 120
The linear equations with one variable are equations that are written as \(ax + b = 0\), where \(a\) and \(b\) are two integers, \(x\) is a variable, and there is only one solution. For example, \(2x + 3 = 8\) is a linear equation with only one variable. As a result, there is only one solution to this equation, which is \(\)x = \frac{5}{2}\(\).
Linear equations can be written in three different ways: point-slope form, standard form, and slope-intercept form.
The equation of a line is given in the image below:
For a straight line, a linear equation has a single variable. The only power of the variable is 1.