Understanding Linear Equations
A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. Linear equations are called “linear” because they represent straight lines when graphed on a coordinate plane.
The general form of a linear equation in one variable is:
$$ ax + b = 0 $$
where:
- ( a ) and ( b ) are constants,
- ( x ) is the variable.
In two variables, a linear equation is represented as:
$$ ax + by = c $$
where:
- ( a ), ( b ), and ( c ) are constants,
- ( x ) and ( y ) are variables.
Types of Linear Equations
-
Linear Equations in One Variable: These equations involve only one variable and can be solved to get a single value for that variable.
- Example: ( 3x + 5 = 11 )
-
Linear Equations in Two Variables: These equations involve two variables and can have multiple solutions that form a line when plotted on a graph.
- Example: ( 2x + 3y = 6 )
Solving Linear Equations in One Variable
To solve linear equations, the goal is to isolate the variable on one side of the equation. Here’s a simple approach:
- Move all variable terms to one side of the equation.
- Move constant terms to the opposite side.
- Isolate the variable by dividing or multiplying if necessary.
Example Problems and Solutions
Problem 1: Linear Equation in One Variable
Solve for ( x ):
$$ 3x – 7 = 11 $$
Solution:
- Move the constant term to the right: $$ 3x = 11 + 7 $$
- Simplify: $$ 3x = 18 $$
- Divide by 3 to isolate ( x ): $$ x = frac{18}{3} = 6 $$
Answer: ( x = 6 )
Problem 2: Linear Equation in Two Variables
Solve the system of equations:
$$ 2x + 3y = 12 $$ $$ x – y = 2 $$
Solution: We can solve this using substitution or elimination. Let’s use substitution.
- Solve the second equation for ( x ): $$ x = y + 2 $$
- Substitute ( x = y + 2 ) into the first equation: $$ 2(y + 2) + 3y = 12 $$
- Distribute and simplify: $$ 2y + 4 + 3y = 12 $$ $$ 5y = 8 $$ $$ y = frac{8}{5} = 1.6 $$
- Substitute ( y = 1.6 ) back into ( x = y + 2 ): $$ x = 1.6 + 2 = 3.6 $$
Answer: ( x = 3.6 ), ( y = 1.6 )
Problem 3: Word Problem Involving Linear Equations
Problem: A school has 5 times as many students as teachers. If the total number of students and teachers is 600, how many students and teachers are there?
Solution:
- Let ( x ) be the number of teachers.
- Then, the number of students is ( 5x ).
- According to the problem: $$ x + 5x = 600 $$
- Combine like terms: $$ 6x = 600 $$
- Divide by 6: $$ x = frac{600}{6} = 100 $$
So, there are ( 100 ) teachers and ( 5 times 100 = 500 ) students.
Answer: 100 teachers and 500 students.
Graphing Linear Equations
Graphing linear equations in two variables (e.g., ( y = 2x + 3 )) helps visualize solutions as points on a line. The slope-intercept form ( y = mx + c ) is often used, where:
- ( m ) is the slope (rise over run),
- ( c ) is the y-intercept.
For example, to graph ( y = 2x + 3 ):
- Start at the y-intercept (0, 3).
- Use the slope ( m = 2 ) (rise 2 units, run 1 unit right) to plot additional points.
Conclusion
Linear equations form the foundation of algebra and have broad applications in problem-solving, graphing, and real-world situations. Mastering these equations helps with more advanced topics and builds a strong mathematical foundation.