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 $\frac{y}{x}$). |

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

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

**Example:**$5x - 2y$ = $7$ and $-26x - 15y$ = $12$ are linear equations in two variables.

**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.

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.

### Solved Example

**Question:**Find the value of x for the given equation below:

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

**Solution:**

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

- 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.

### 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:**

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:**

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).

### 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:**

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.