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PREPINSTA PRIME

# 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 :

- Standard Form
- Slope-Intercept Form
- Point-Slope Form

**1. Standard Form**

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

##### Ax + By = C

Where:

- $A$ and $B$ are coefficients (constants) representing the coefficients of $x$ and $y$ terms, respectively.
- $C$ is a constant term.

The standard form requires that $A$ and $B$ are both integers and that $A$ is non-negative. Also, $A$ and $B$ 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:

##### $y=mx+b$

Where:

- $m$ is the slope of the line, representing the rate of change between $y$ and $x$.
- $b$ is the y-intercept, which is the value of $y$ when $x$ 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:

##### $y−y_{1}=m(x−x_{1})$

Where:

- $m$ is the slope of the line, as explained in the slope-intercept form.
- $(x_{1},y_{1})$ 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_{2 }

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_{2 }

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: a
_{1}x + b_{1}y = c_{1}and a_{2}x + b_{2}y = c_{2}

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) $y=mx+b$

b) $y=mx−b$

c) $y=bx+m$

d) $y=bx−m$

**Answer :** (a)

**Explanation: **

The correct answer is (a). The slope-intercept form of a linear equation is $y=mx+b$, where $m$ represents the slope and $b$ 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

**Answer:** (a)

**Explanation:** The correct answer is (a). The point-slope form of a linear equation is written as $y−y_{1}=m(x−x_{1})$, where $(x_{1},y_{1})$ represents the coordinates of a point on the line, and $m$ is the slope.

**Question 3:**

What does the standard form of a linear equation look like? a) $y=mx+b$

b) $y=bx+m$

c) $Ax+By=C$

d) $Ax+By=D$

Explanation: The correct answer is (c). The standard form of a linear equation is $Ax+By=C$, where $A$ and $B$ are coefficients representing the coefficients of $x$ and $y$, respectively, and $C$ is a constant term.

**Question 4: **

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

a) $y=mx+b$

b) $y−y_{1}=m(x−x_{1})$

c) $y=Ax+By$

d) $y=m1 x+b$

Explanation: The correct answer is (b). The point-slope form of a linear equation is represented as $y−y_{1}=m(x−x_{1})$, where $(x_{1},y_{1})$ represents the coordinates of a point on the line, and $m$ 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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