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Linear Equations

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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

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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.

Example: $y$ = $5x + 2,\ 7x$ = $\frac{3}{2}$ $y + 2$, $\frac{y}{4}$ = $8 - 5x$

Linear Equations in One Variable

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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. 

Example: $7x + 9$ = $0,\ 2x + 5$ = $-5x + 8$.

Linear Equations in Two Variables

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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.

Example: $5x - 2y$ = $7$ 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.

No Solution: If two lines are parallel to each other, then they will never intersect each other. So, the system has no solution.

Infinite Solution: If two lines lie on top of each other, then we can say that there are infinite number of solutions. 

Slope-Intercept Form

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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.

Point–Slope Form

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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.

Linear Equation with Fractions

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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.

Solved Example

Question: Find the value of x for the given equation below:
-12x - $\frac{4}{5}$ = $\frac{2}{7}$
Solution:
Given equation is -12x - $\frac{4}{5}$ = $\frac{2}{7}$

The LCM of 5 and 7 is 35.

35 (-12x - $\frac{4}{5}$) = 35 ($\frac{2}{7}$)

- 420x - 28  = 10
- 420x = 38
x = 0.09

Steps to Solve Linear Equation

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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.

Linear Equations Examples 

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Given below are some of the examples on linear equations.

Solved Examples

Question 1: Write the equation of a straight line in standard form which passes through the point (2, 3) and has slope 5.
Solution:
We know that equation of a line that passes through the point (2, 3) and has slope 5 can be obtained by the slope point form.
Equation of slope point form is y - $y_{1}$ = m (x - $x_{1}$).
Here. m = 5 and x$_{1}$ = 2 and y$_{1}$ = 3
It implies that y - 3 = 5(x - 2)
By comparing with the standard form, Ax + By = C, it can be written as 5x - y = 7.
We get A = 5, B = -1 , C = 7.

Question 2: Write the equation of a straight line 4x + 3y = 5 in point slope form.
Solution:
Standard form of a straight line is 4x + 3y = 5
Let us convert the given equation in slope point form
3y = 5 - 4x

y = $\frac{1}{3}$ (5 - 4x) 

The above equation can be rewritten in point slope form as
y = -$\frac{4}{3}$ (x - $\frac{5}{4}$)

The above equation is the point slope form of a line having slope -$\frac{4}{3}$ and passing through the point (-$\frac{5}{4}$, 0).

Linear Equation Word Problems

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

Solved Examples

Question 1: Find the number, if 262 is added to a number, the result is 85 more than 4 times the number.
Solution:
Step 1: Let 'x' be the number we wish to find.
According to the problem, the statement in mathematical form can be written as
262 + x = 4x + 85
177 = 3x
Therefore, x = 59.

Step 2: Let 'x' be the number we wish to find.
According to the problem, the statement in mathematical form can be written as
262 + x = 4x + 85
177 = 3x
Therefore, x = 59.

Question 2: Find the cost for each adult ticket if the equations are 10x - 4y = 75 and 5x - 4y = 40, represent the money collected from music concert tickets sales during two class periods, where, 'x' represents the cost for each adult ticket and 'y' represents the cost for each student ticket.
Solution:
Since 'x' be the cost for each adult ticket, to solve this problem, we need to find the value of x.
Solve the two given equations simultaneously
10x - 4y = 75
5x - 4y =  40
We get 5x = 35
x = 7
Therefore, the cost of each adult ticket is 7.