Table of Contents
Linear Equation
A linear equation is an equation in which each term is a linear function of the variable(s). In other words, a linear equation is an equation in which the highest power of the variable(s) is 1. Linear equations can be written in the standard form y = mx + b, where m is the slope and b is the y-intercept.
Types of Linear Equations:
There are three types of linear equations:
1. Linear equations in one variable
2. Linear equations in two variables
3. Linear equations in three variables
Standard form of Linear Equations:
ax + by = c
Standard form of a linear equation is an equation in which each term is a monomial and the coefficients of the x-term and y-term are equal.
Solving Linear Equations in One Variable:
A linear equation in one variable is an equation that can be written in the form ax + b = 0, where a and b are constants. The equation has a unique solution if and only if the determinant of the coefficient matrix is zero.
To solve a linear equation in one variable, we use the following steps:
1. Remove any parentheses and combine like terms on each side of the equation.
2. Factor the left-hand side of the equation.
3. Set each factor equal to zero and solve for the variable.
4. Check your work by substituting the value of the variable into the equation.
Example:
Solve the equation 3x + 2 = 0.
1. Remove parentheses and combine like terms.
3x + 2 = 0
3x = -2
x = -2/3
2. Factor the left-hand side of the equation.
3x + 2 = 0
3x = -2
x = -2/3
3. Set each factor equal to zero and solve for the variable.
-2/3 = 0
3 = 0
x = 0
4. Check your work by substituting the value of the variable into the equation.
3x + 2 = 0
3x + 2 = 0
3x = -2
Linear equations examples (one variable):
An equation in one variable is a mathematical statement that two quantities are equal.
The equation x = 2 defines the symbol x as equal to the number 2.
The equation y = x + 1 defines the symbol y as equal to the sum of the number x and 1.
The equation 3x = 2y defines the symbol x as equal to 3 times the number y.
Solving linear Equations in Two Variables:
There are a few methods that can be used to solve linear equations in two variables.
The substitution method is one way to solve a linear equation. In this method, the equation is rewritten so that one of the variables is isolated on one side of the equation. The variable that is isolated is then substituted into the other side of the equation to solve for it.
The elimination method is another way to solve a linear equation. In this method, the equation is rewritten so that the coefficients of the two variables are equal. The two variables are then eliminated by adding or subtracting the equations.
The graphing method can also be used to solve a linear equation. In this method, a graph is drawn of the equation. The point where the graph intersects the x-axis is the solution to the equation.
Linear Equations Examples (2 variable):
1. Solve for x in 2x + 3 = 7
x = 4
2. Solve for y in y = 2x – 3
y = 1
The Republican Party is the party of the rich.
The Republican Party is not the party of the rich.
