Linear Equations

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Sub Topics Standard Form of a Linear Equation Linear Equations in One Variable Linear Equations in Two Variables Slope-Intercept Form Point–Slope Form Linear Equation with Fractions Steps to Solve Linear Equation Linear Equations Examples  Linear Equation Word Problems

Linear equation is an algebraic equation which may consist of a single variable and constant or product of two or more constants. It may consist of more than one variable. It can be solved for the values of the variable. By term solving linear equation, we mean that we are going to find the value of variable in the given equation. After determining value of one variable, we put this value in linear equation, we observe that the linear equation satisfies the given linear equation. We also call it the solution to the given linear equation. A linear expression is basically a mathematical statement that performs mathematical operations. The variables which are used in the linear equation are unknown and they can never be in one of the following forms: The variable used in a linear equation cannot have an exponent. The variable must not be found under Square root or any other root. The variables say x, y; must not multiply and divide each other (i.e. $\frac{x}{y}$ or $\frac{y}{x}$).

Standard form of a linear equation in two variables x and y is given as
Ax + By = C
Where, A, B and C are integers and have no common factor other than 1.

Linear equation is an equation for a straight line.
Examples: y = 5x + 2, 7x = $\frac{3}{2}$y + 2, $\frac{y}{4}$ = 8 - 5x Linear equation with one variable consists of numbers or constants and multiples of a variable. The standard form of linear equation with one variable is ax + b = 0, where, a and b are constants and x is the variable.
Examples: 7x + 9 = 0, 2x + 5 = -5x + 8.
Linear equations in two variables is an equation that can be written in the form ax + by + c = 0, where a, b and c are non zero real numbers.
Examples: 5x - 2y = 7 and -26x - 15y = 12 are linear equations in two variables.

While solving linear systems with two variables, there are three possibilities of solutions:

1. One solution
2. No solution
3. Infinite solution

One Solution: If system given in two variable has one solution, then it is an ordered pair and that ordered pair is a solution of both equations.
Infinite Solution: If two lines lie on top of each other, then we can say that there are infinite number of solutions.
The slope intercept form is the equation of a straight line in the form: y = mx + c, where, m is the known as the slope of the line with the x-axis and c is known as the intercept on y-axis.

If the intercept is given on x-axis, then the slope intercept form of the straight line can be written as y = m(x - d), where m is the slope of the line and d is the intercept of the line on x- axis. The point slope form of a straight line is the equation of line having slope 'm' and passing through the point ($x_{1}$, y$_{1}$).
Equation for point-slope form is y - y$_{1}$ = m(x - x$_{1}$)
where, $(x_{1}, y_{1})$: Known point.
m: Slope of the line.
(x, y): Other point on the line.
To find other points on the line, one should know one point on the line and the slope of the line.
When linear equations contain fractions, it is better to clear fractions by finding the lowest common multiple (lcm) of all the fractions and then multiply every term in the equation by the LCM. Given below are the steps to solve linear equations:

• For the given problem, define the variables you wish to find.
• Create equation based on the given statements in the problem.
• Combine the like terms and isolate the variable.
• Distributive property can be used to remove symbols of grouping.
• Fractions can be eliminated by multiplying each term on both sides by the lowest common denominator.
• Using algebraic methods, solve your system and check whether the answer is reasonable.
• Verify the answer with the conditions given in the problem.

Given below are some of the examples on linear equations. Given below are some of the word problems on linear equations.