Formulas for Linear Equation Problems

August 3, 2023

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:

$y = m x + b$

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 p₂

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 q₂

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) $y = m x + b$

b) $y = m x - b$

c) $y = b x + m$

d) $y = b x - m$

Answer : (a)

Explanation:

The correct answer is (a). The slope-intercept form of a linear equation is $y = m x + b$ , 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) $y = m x + b$

b) $y = b x + m$

c) $A x + B y = C$

d) $A x + B y = D$

Explanation: The correct answer is (c). The standard form of a linear equation is $A x + B y = C$ , 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) $y = m x + b$

b) $y - y_{1} = m (x - x_{1})$

c) $y = A x + B y$

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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Formulas for Linear Equation Problems

Introduction to Formulas of Linear Equations

Formulas & Definitions for Linear Equations

Forms of Linear Equations

1. Standard Form

Ax + By = C

2. Slope-Intercept Form

$y = m x + b$

3. Point-Slope Form

$y - y_{1} = m (x - x_{1})$

Linear equations in one variable

Linear equations in two variable

Linear equations in three variable

Formulas and Methods to solve Linear equations

Important Formulas of Linear Equation & key points to Remember

Questions on Formulas of Linear Equation:

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Formulas for Linear Equation Problems

Introduction to Formulas of Linear Equations

Formulas & Definitions for Linear Equations

Forms of Linear Equations

1. Standard Form

Ax + By = C

2. Slope-Intercept Form

y=mx+b

3. Point-Slope Form

y−y1​=m(x−x1​)

Linear equations in one variable

Linear equations in two variable

Linear equations in three variable

Formulas and Methods to solve Linear equations

Important Formulas of Linear Equation & key points to Remember

Questions on Formulas of Linear Equation:

Prime Course Trailer

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Also Check Out

Get over 200+ course One Subscription

Checkout list of all the video courses in PrepInsta Prime Subscription

Checkout list of all the video courses in PrepInsta Prime Subscription

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Unlock this article for Free,
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$y = m x + b$

$y - y_{1} = m (x - x_{1})$