Linear Equations Formula

A linear equation is known as an algebraic equation that represents a straight line. It is composed of variables and constants. Linear equations consist of the first-order, which involves the highest power to any of the involved variables i.e.

• It is also considered as the polynomial of a degree
• The equation that contains only one variable is known as the homogeneous equation.

The formula used to represent the linear equations is called the linear equation formula. There are various ways to represent the linear equations such as,

Standard Form

The most generalized form of a linear equation in two variables is:

Ax + By = C where A, B, and C are constants, and x and y are variables.

This form is particularly useful for quickly determining the intercepts and is common in algebraic problems.

Example: Convert the equation y = 2/3x - 4 into standard form.

Solution:

We start with the given equation: y = 2/3x − 4
Multiply everything by 3 to eliminate the fraction: 3y = 2x − 12

Rearrange to get it in Ax + By = C form: 2x − 3y = 12

Read More: Standard Form

Slope-Intercept Form

This is another highly popular form for representing linear equations, especially when graphing:

y = mx + by where m is the slope of the line, and b is the y-intercept.

This form directly shows how y changes with x and where the line crosses the y-axis.

Example: Find the equation of a line with slope m = −5m and y-intercept b = 7.

Solution:

Since we are given the slope and y-intercept, we directly substitute into the equation:
y = −5x + 7

Read More: Slope-Intercept Form

Point-Slope Form

Useful for when you know a point on the line (x₁, y₁) and the slope

m: y − y₁ = m(x − x₁)

This form is handy for writing the equation of a line when you are given a point on the line and the slope.

Example: Find the equation of the line passing through (3, -2) with a slope of 4.

Solution:

Using the point-slope formula: y − (−2) = 4(x − 3)

Simplify: y + 2 = 4x − 12

Subtract 2 from both sides:

y = 4x − 14 (which is also in slope-intercept form)

Read More: Point-Slope Form

Intercept Form

If the line intercepts the axes at (a, 0) and (0, b), then the equation can be expressed as:

\frac{x}{a} + \frac{y}{b} = 1

This is useful when you know where the line crosses the x-axis and y-axis.

Example: Find the equation of the line that intercepts the x-axis at (6,0) and the y-axis at (0, 4).

Solution:

Using the formula: \frac{x}{a} + \frac{y}{b} = 1

Substituting a = 6 and b = 4:

\frac{x}{6} + \frac{y}{4} = 1

Linear Equation in One Variables

A linear equation in one variable takes the form ax + b = 0 or equivalently ax = b, where:

• x is the variable.
• a and b are constants with a ≠ 0 to ensure the equation is linear and not trivial.

Solving Linear Equations in One Variable

The goal is to isolate the variable x to find its value. The steps are simplified to make the process clear:

• Step 1: Simplify the equation. If the equation includes fractions, multiply through by the least common denominator to eliminate them.
• Step 2: Rearrange the equation. Move all terms containing x to one side and constants to the other to isolate x.
• Step 3: Solve for x by dividing both sides by the coefficient a.

Examples of linear equations in one variable

• 22x = 65
• 6/5 + 1/3 x = 2
• 8y – 3 = 7
• 4/3 (z - 2) = 0

After analyzing these examples we get to know that each one of the equations has only one variable in it and the highest power the variable gets is 1. These algebraic equations can be solved by taking all the variables on the left-hand side (L.H.S) and constants on the right-hand side (R.H.S), to solve the corresponding variable value

Linear equation in Two Variable

A linear equation with two variables is represented as ax + by + c = 0, where:

• x and y are variables,
• a, b, and c are constants,
• a and b should not be zero to ensure the equation is not degenerate.

Example:

Consider a cricket match where two Indian batsmen scored a total of 158 runs together. We can express their individual scores as x and y. The equation that models this situation is: x + y = 158

Solving Linear Equation in Two Variable

Linear equations in two variables can have infinitely many solutions. To find solutions, you choose values for one variable and solve for the other.

Step-by-Step Solution:

Example: Solve x + 3y = 6

• Assume x = 3, then solve for y:
  • 3 + 3y = 6
  • 3y = 3
  • y = 1
• Verify:
  • 3 + 3 × 1 = 6 confirms the solution is correct.
• Find another solution by assuming a different value for x:
  • Assume x = 6:
  • 6 + 3y = 6
  • 3y = 0
  • y = 0

Infinite Solutions

By varying x and solving for y, or vice versa, you can find numerous solutions, demonstrating the concept of infinite solutions for linear equations in two variables.

Related Reads:

• Linear Equations in One Variable
• Solve Linear Equations with variables on both sides

Solved Examples of Linear Equations Formula

Question 1: Solve for y, 6y – 3 = 0

Solution:

Solving for the value of y,

Adding 3 to both sides of the equation,

⇒ 6y - 3 + 3 = 3
⇒ 6y = 3

Dividing both sides of the equation by 6
⇒ y = 3/6

Simplifying the equation,
⇒ y = 1/2

Question 2: Solve the equation in x, 4/5x -5 = 15

Solution:

4/5x - 5 = 15

Taking constants to RHS,

4/5x = 15 + 5
4/5x = 20
x = 100/4
x = 25

Question 3: There are two numbers, one equal to 7/6 and the other equal to 1/3 times some number x. The sum of these two numbers is 1. Find x.

Solution:

The sum of both the numbers is 1 so the equation will be, 7/6 + 1/3x = 1

Taking all the constants to the R.H.S of the equation.

1/3x = 1 - 7/6
1/3x = -1/6

Multiplying both the side of the equation by 3
3 (1/3x) = 3 × (-1/6)
x = -1/3

Question 4: Solve the equation in x, 3x + 5y = 33, where y = 3

Solution:

We have been provided with an equation 3x + 5y = 33

We have to find the value of x as the value of y is provided in the question y= 3

So, putting the value of y in the equation
3x + 5(3) = 33
3x + 15 = 33

By taking all the constants to the R.H.S of the equation.
3x = 33 - 15
3x = 18
x= 18/3
x = 6

So here the value of x is 6

Question 5: There are two numbers, one equal to 2/4 some number y, and the other equal to 1/3 times some number x. The sum of these two numbers is 3. Find y. And the value of x is x = 2

Solution:

The sum of both the numbers is 3 so the equation will be, 2/4y + 1/3x= 3

Pitting the value of x the equation will be

2/4y + 1/3(2)= 3
2/4y + 2/3= 3

Taking all the constants to the R.H.S of the equation.

2/4y = 3-2/3
2/4y = 4/3
y= 4*4/3*2
y = 16/6
y = 8/3

So, the value of y will be 8/3

Linear Equations Formula - Practice Problems

Question 1: Solve for x: 5x + 7 = 12.

Question 2: Solve for y: 9y − 4 = 23.

Question 3: Solve for z: 3z + 11 = 2z + 7.

Question 4: Solve for x: 7x − 5 = 2x + 20.

Question 5: Solve for y: 12y + 4 = 8y − 16.

Question 6: Solve for y: \frac{5}{2}z - 3 = 7 .

Question 7: Solve for x: 4x + \frac{3}{5} = \frac{2}{3} .

Question 8: Solve for y: \frac{3}{4}y + 7 = 2.

Question 9: Solve for x: \frac{2}{3}x - 5 = \frac{1}{4}.

Question 10: Solve for z: \frac{6}{7}z + 3 = 10.