# Formulas for Linear Equation Problems

## Introduction to Formulas of Linear Equations

A linear equation is also known as an algebraic equation in which each term has an exponent of one. The graph representation of the equation shows a straight line. Standard form of linear equation is y = m x + b. Where, x is the variable and y, m, and b are the constants. On this page you will find complete Formulas for Linear Equation that will help you to solve questions of all levels.

### Formulas & Definitions for Linear Equations

• A linear equation is an algebraic equation in which each term has an exponent of one and the graphing of the equation results in a straight line.
• Standard form of linear equation is y = mx + b. Where, x is the variable and y, m, and b are the constants.

### Forms of Linear Equations

There are mainly 3 forms of Linear Equation :

1. Standard Form
2. Slope-Intercept Form
3. Point-Slope Form

#### 1. Standard Form

The standard form of a linear equation is typically written as:

##### Ax + By = C

Where:

• and are coefficients (constants) representing the coefficients of and terms, respectively.
• is a constant term.

The standard form requires that and are both integers and that is non-negative. Also, and should not have any common factors other than 1. This form is commonly used in algebraic manipulation and solving systems of linear equations.

#### 2. Slope-Intercept Form

The slope-intercept form of a linear equation is written as:

Where:

• is the slope of the line, representing the rate of change between and .
• is the y-intercept, which is the value of when is equal to 0. It represents the point where the line intersects the y-axis.

This form is particularly useful for graphing linear equations and quickly identifying the slope and y-intercept of the line.

#### 3. Point-Slope Form

The point-slope form of a linear equation is given by:

Where:

• is the slope of the line, as explained in the slope-intercept form.
• (�1,�1) represents the coordinates of a point on the line.

This form is useful when you know a specific point on the line and its slope, allowing you to write the equation directly without having to calculate the y-intercept.

### Linear equations in one variable

• A Linear Equation in one variable is defined as ax + b = 0
• Where, a and b are constant, a ≠ 0, and x is an unknown variable
• The solution of the equation ax + b = 0 is x =$– \frac{b}{a}$ . We can also say that $– \frac{b}{a}$ is the root of the linear equation ax + b = 0.

### Linear equations in two variable

• A Linear Equation in two variables is defined as ax + by + c = 0
• Where a, b, and c are constants and also, both a and b ≠ 0

### Linear equations in three variable

• A Linear Equation in three variables is defined as ax + by + cz = d
• Where a, b, c, and d are constants and also, a, b and c ≠ 0

### Formulas and Methods to solve Linear equations

• Substitution Method

Step 1:   Solve one of the equations either for x or y.

Step 2:    Substitute the solution from step 1 into the other equation.

Step 3:    Now solve this equation for the second variable.

• Elimination Method

Step 1:    Multiply both the equations with such numbers to make the coefficients of one of the two unknowns numerically same.

Step 2:    Subtract the second equation from the first equation.

Step 3:    In either of the two equations, substitute the value of the unknown variable. So, by solving the equation, the value of the other unknown variable is obtained.

• Cross-Multiplication Method

Suppose there are two equation,

$p_{1}x +q_{1}y = r_{1}$  ……..(1)

$p_{2}x +q_{2}y = r_{2}$  ……..(2)

Multiply Equation (1) with p

Multiply Equation (2) with p1

$p_{1}p_{2}x +q_{1}p_{2}y = r_{1}p_{2}$

$p_{1}p_{2}x +p_{1}q_{2}y = p_{1}r_{2}$

Subtracting,
$q_{1}p_{2}y – p_{1}q_{2}y = r_{1}p_{2} – p_{1}r_{2}$

or,  y (q₁ p₂ – q₂p₁) = r₂p₁ – r₁p₂

Therefore, y = $\frac{r_{2}p_{1} – r_{1}p_{2} }{q_{1}p_{2} – q_{2}p_{1} }$

= $\frac{r_{1}p_{2} – r_{2}p_{1} }{q_{2}p_{1} – q_{1}p_{2} }$

where (p₁q₂ – p₂q₁) ≠ 0

Therefore, $\frac{y}{r_{1}p_{2} – r_{2}p_{1} } = \frac{1}{q_{2}p_{1} – q_{1}p_{2} }$        …(3)

Multiply Equation (1) with q

Multiply Equation (2) with q1

$p_{1}q_{2}x +q_{1}q_{2}y = r_{1}q_{2}$

$q_{1}p_{2}x +q_{1}q_{2}y = q_{1}r_{2}$

Subtracting,

$p_{1}q_{2}x – p_{2}q_{1}x = r_{1}q_{2} – q_{1}r_{2}$

or , x(p₁q₂ – p₂q₁) = (q₁r₂ – q₂r₁)

or,  x = $\frac{q_{1}r_{2} – r_{1}q_{2}}{p_{1}q_{2} – p_{2}q_{1}}$

Therefore, $\frac{x}{q_{1}r_{2} – r_{1}q_{2}} = \frac{1}{p_{1}q_{2} – p_{2}q_{1}}$   … (4)

where (p₁q₂ – p₂q₁) ≠ 0

From equations (3) and (4), we get,

$\frac{x}{q_{1}r_{2} – r_{1}q_{2}} =\frac{y}{r_{1}p_{2} – p_{1}r_{2}} = \frac{1}{p_{1}q_{2} – p_{2}q_{1}}$

where (p₁q₂ – p₂q₁) ≠ 0

Note: Shortcut to solve this equation will be written as

$\frac{x}{q_{1}r_{2} – r_{1}q_{2}} =\frac{y}{r_{1}p_{2} – r_{2}p_{1} } = \frac{1}{q_{2}p_{1} – q_{1}p_{2} }$

which means,

x =$\frac{q_{1}r_{2} – r_{1}q_{2}}{q_{2}p_{1} – q_{1}p_{2} }$

y = $\frac{r_{1}p_{2} – r_{2}p_{1}}{q_{2}p_{1} – q_{1}p_{2}}$

### Important Formulas of Linear Equation & key points to Remember

• Suppose, there are two linear equations: a1x + b1y = c1 and a2x + b2y = c2

Then,

(A) If $\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}$, then there will be one solution, and the graphs will have intersecting lines.

(B) If$\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}}$, then there will be numerous solutions, and the graphs will have coincident lines.

(C) If $\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}≠ \frac{c_{1}}{c_{2}}$ then there will be no solution, and the graphs will have parallel lines.

### Questions on Formulas of Linear Equation:

Question 1:

What is the slope-intercept form of the equation of a line?

a)

b)

c)

d)

Explanation:

The correct answer is (a). The slope-intercept form of a linear equation is , where represents the slope and represents the y-intercept.

Question 2:

Which form of a linear equation is useful when you know a specific point on the line and its slope?

a) Point-Slope Form

b) Slope-Intercept Form

c) Standard Form

d) None of the above

Explanation: The correct answer is (a). The point-slope form of a linear equation is written as , where (1,1) represents the coordinates of a point on the line, and is the slope.

Question 3:

What does the standard form of a linear equation look like? a)

b)

c)

d)

Explanation: The correct answer is (c). The standard form of a linear equation is , where and are coefficients representing the coefficients of and , respectively, and is a constant term.

Question 4:

Which of the following is the correct representation of the point-slope form of a linear equation?

a) �=��+�

b) �−�1=�(�−�1)

c) �=��+��

d)

Explanation: The correct answer is (b). The point-slope form of a linear equation is represented as �−�1=�(�−�1), where (�1,�1) represents the coordinates of a point on the line, and is the slope of the line. This form is useful when you know a specific point on the line and its slope.

Question 5:

How can you eliminate fractions from a linear equation?

a) Multiply both sides of the equation by a common denominator

b) Divide both sides of the equation by a common denominator

c) Add both sides of the equation by a common denominator

d) Subtract both sides of the equation by a common denominator

Explanation: The correct answer is (a). To eliminate fractions from a linear equation, multiply both sides of the equation by a common denominator. This process will clear the fractions and make the equation easier to solve.

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