- Math Article

## Solving Linear Equations

## What Does Solving Linear Equations Mean?

## How to Solve Linear Equations?

- Graphical Method
- Elimination Method
- Substitution Method
- Cross Multiplication Method
- Matrix Method
- Determinants Method

## Graphical Method of Solving Linear Equations

Check: Graphical Method of Solving Linear Programming

The graph of 2x + 3y = 9 and x – y = 3 will be as follows:

## Elimination Method of Solving Linear Equations

Consider the same equations as

So, multiply equation (ii) × 2 and then subtract equation (i)

Now, put the value of y = 0.6 in equation (ii).

In this way, the value of x, y is found to be 3.6 and 0.6.

## Substitution Method of Solving Linear Equations

Now, consider equation (ii) and isolate the variable “x”.

Now, substitute the value of x in equation (i). So, equation (i) will be-

Now, substitute “y” value in equation (ii).

## Cross Multiplication Method of Solving Linear Equations

x /(b 1 c 2 − b 2 c 1 ) = y / (c 1 a 2 − c 2 a 1 ) = 1 /(b 2 a 1 − b 1 a 2 )

For example, consider the equations

Now, solve using the aforementioned formula.

x = (b 1 c 2 − b 2 c 1 ) / (b 2 a 1 − b 1 a 2 )

Putting the respective value we get,

y = (c 1 a 2 − c 2 a 1 ) / (b 2 a 1 − b 1 a 2 )

## Matrix Method of Solving Linear Equations

These equations can be written as:

Here, the A matrix, B matrix and X matrix are:

Now, multiply (i) by A -1 to get:

A −1 AX = A −1 B ⇒ I.X = A −1 B

## Determinant Method of Solving Linear Equations (Cramer’s Rule)

For Linear Equations in Two Variables:

For Linear Equations in Three Variables:

## Related Video: Solving an Equation

## Methods of Solving Linear Equations in One Variable

For example, consider the equation 2x + 4 + 7 = 4x – 3 + x

Here, combine the “x” terms and bring them on one side.

## Methods of Solving Linear Equations in Two Variables

## Methods of Solving Linear Equations in Three or More Variables

Check: Solve Linear Equation in Two Or Three Variables

## Topics Related to Solution of Linear Equations:

Frequently asked questions, what is a linear equation.

## What are the Methods of Solving Linear Equations?

The 6 most common methods of solving a linear equation are:

## How to Solve Linear Equations with Fractions?

To solve a linear equation with fraction, follow these steps:

- Step 1: Make any complex fraction into a simple fraction
- Step 2: Find the LCM of all denominators
- Step 3: Multiply the equation with the LCM of the denominator
- Step 4: Cancel out the fractions as all the denominators can be divided by the LCM value
- Step 5: Solve the final linear equation using any of the methods explained here

Learn more about fraction here .

Put your understanding of this concept to test by answering a few MCQs. Click ‘Start Quiz’ to begin!

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Plotting the ordered pair solutions and drawing the line:

Practice Problem 1a: Solve the system by graphing.

Practice Problem 2a: Solve the system by the substitution method.

Practice Problem 3a: Solve the system by the elimination method.

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## Solving Linear Equations

## Solving Linear Equations in One Variable

4x - 8x = -10 - 8 -4x = -18 4x = 18 x = 18/4 On simplifying, we get x = 9/2.

## Solving Linear Equations by Substitution Method

x = 6 - y Substituting the value of 'x' in 2x + 4y = 20, we get,

## Solving Linear Equations by Elimination Method

2(x) + 2(3y) = 2(18). Now equation (2) becomes, 2x + 6y = 36 -----------(2)

Therefore, by solving linear equations, we get the value of x = 3 and y = 5.

## Graphical Method of Solving Linear Equations

Let us take some values for 'x' and find the values for 'y' in the equation y = x + 2.

Plotting these points on the coordinate plane, we get a graph like this.

## Cross Multiplication Method of Solving Linear Equations

## Topics Related to Solving Linear Equations

Check the given articles related to solving linear equations.

- Linear Equations
- Application of Linear Equations
- Two-Variable Linear Equations
- Linear Equations and Half Planes
- One Variable Linear Equations and Inequations

## Solving Linear Equations Examples

Example 1: Solve the following linear equations by the substitution method.

3x + y = 13 --------- (1) 2x + 3y = 18 -------- (2)

Therefore, by the substitution method, the value of x is 3 and y is 4.

Example 2: Using the elimination method of solving linear equations find the values of 'x' and 'y'.

3x + y = 21 ------ (1) 2x + 3y = 28 -------- (2)

x + 2y - 16 = 0 --------- (1) 4x - y - 10 = 0 ---------- (2)

\(a_{1}\) = 1, \(a_{2}\) = 4, \(b_{1}\) = 2, \(b_{2}\) = -1, \(c_{1}\) = -16, and \(c_{2}\) = -10.

By cross multiplication method,

Substituting the values in the formula we get,

go to slide go to slide go to slide

## Practice Questions on Solving Linear Equations

Faqs on solving linear equations, what does it mean by solving linear equations.

## How to Use the Substitution Method for Solving Linear Equations?

## How to Use the Elimination Method for Solving Linear Equations?

## What is the Graphical Method of Solving Linear Equations?

## What are the Steps of Solving Linear Equations that has One Variable?

## What are the Steps of Solving Linear Equations having Three Variables?

## What are the 4 Methods of Solving Linear Equations?

The methods for solving linear equations are given below:

## Solving Systems of Linear Equations

A system of linear equations is just a set of two or more linear equations.

There are three possibilities:

- The lines intersect at zero points. (The lines are parallel.)
- The lines intersect at exactly one point. (Most cases.)
- The lines intersect at infinitely many points. (The two equations represent the same line.)

There are a few different methods of solving systems of linear equations:

- The Graphing Method . This is useful when you just need a rough answer, or you're pretty sure the intersection happens at integer coordinates. Just graph the two lines, and see where they intersect!
- The Substitution Method . First, solve one linear equation for y in terms of x . Then substitute that expression for y in the other linear equation. You'll get an equation in x . Solve this, and you have the x -coordinate of the intersection. Then plug in x to either equation to find the corresponding y -coordinate. (If it's easier, you can start by solving an equation for x in terms of y , also – same difference!)

Solve the system { 3 x + 2 y = 16 7 x + y = 19

Substitute 19 − 7 x for y in the first equation and solve for x .

3 x + 2 ( 19 − 7 x ) = 16 3 x + 38 − 14 x = 16 − 11 x = − 22 x = 2

Substitute 2 for x in y = 19 − 7 x and solve for y .

Solve the system { 4 x + 3 y = − 2 8 x − 2 y = 12

− 8 x − 6 y = 4 8 x − 2 y = 12 _ − 8 y = 16

Substitute for y in either of the original equations and solve for x .

4 x + 3 ( − 2 ) = − 2 4 x − 6 = − 2 4 x = 4 x = 1

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## The elimination method for solving linear systems

\begin{cases} 3y+2x=6\\ 5y-2x=10 \end{cases}

We can eliminate the \(x\)-variable by addition of the two equations.

\begin{cases} 3y+2x=6 \\ \underline{+\: 5y-2x=10} \end{cases}

$$=8y\: \: \: \: \; \; \; \; =16$$

$$\begin{matrix} \: \: \: y\: \: \: \: \: \; \; \; \; \; =2 \end{matrix}$$

$$3\cdot {\color{green} 2}+2x=6$$

The solution of the linear system is \((0, 2)\).

\begin{cases} 3x+y=9\\ 5x+4y=22 \end{cases}

Begin by multiplying the first equation by \(-4\) so that the coefficients of \(y\) are opposites

\begin{cases}-12x-4y=-36 \\ \underline{+5x+4y=22 }\end{cases}

$$=-7x\: \: \: \: \: \: \: \: \: \: =-14$$

$$\begin{matrix} \: \:\; \:\: x\: \: \: \: \: \: \: \: \: \: \:=2 \end{matrix}$$

Substitute \(x\) in either of the original equations to get the value of \(y\)

$$3\cdot {\color{green} 2}+y=9$$

The solution of the linear system is \((2, 3)\)

## Video lesson

Solve the following linear system using the elimination method

\begin{cases} 2y - 4x = 2 \\ y = -x + 4 \end{cases}

- Properties of exponents
- Scientific notation
- Exponential growth functions
- Monomials and polynomials
- Special products of polynomials
- Polynomial equations in factored form
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- Completing the square
- The quadratic formula
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- The distance and midpoint formulas
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- Division of polynomials
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## Linear Equations

T.1 The same expression representing a real number may be added to both sides of an equation.

Add -x to both sides to obtain

Add -(1/2)x to both sides to obtain

Multiply both sides by 1/2 to obtain

Thus, the solution set of (b) is {5/6} .

Add -b to both sides to obtain

Multiply both sides by 1/a to obtain

The next two examples are of equations that reduce to linear equations.

We expand both sides to obtain

Add -20y^2 to both sides to obtain

We now solve as in the previous examples.

Thus the solution set is {1} .

Example 1. Solve cx-3a=b for x .

This last equation expresses x in terms of the other symbols.

Example 2. Solve 3ay-2b=2cy for y .

Multiply both sides by 1/((3a-2c))

Example 3. Solve a/x+b/(2x)=c for x .

Example 4. Solve A=P(1+rt) for r .

Multiply both sides by 1/(Pt) .

Example 5. Solve 1/R=1/r_1+1/r_2 for r_1 .

Add the two terms on the right—hand side.

6.4 Solving Statement Problems

1. Read the problem carefully until the situation is thoroughly understood.

3. Establish the relationship between the unknown and the other quantities in the problem.

4. Find the information that tells which two quantities are equal.

5. Use the information in (4) to write the equation.

6. Solve the equation and check the solution to see that it satisﬁes the original problem.

Step 2. Let x be the number of dollars that Joe received.

Step 3. Then 3x is the number of dollars that Bob received

Step 4. Bob and Joe together earned $ 60 . Step 5. 3x+x=60

Step 2. Let x be the tens digit.

Step 3. Then 12 - x is the units digit.

Step 4. If the digits are reversed then the number is decreased by 36

Step 5. 10(12-x)+x = 10x+ (12-x) -36

Check. (83+1/3)48+80*50=60(50+83+1/3)

Problems involving velocities (or speeds) will use the formula

Step 2. Let x be the speed on the dirt road.

Step 3. Then x+25 is the speed on the highway.

[(x+25)(mi)/(hr)](4hrs)+[x(mi)/(hr)](3hrs)=380mi

Step 2. Let x be the number of hours that it would take them to cut the lawn Working together.

## IMAGES

## VIDEO

## COMMENTS

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