Want to master in techniques of Simultaneous Linear Equations? Then, this page plays a vital role in your practice session. To help you achieve all your academic goals, we have again come up with this guide on **Introduction to Simultaneous Linear Equations.**

Excited to learn what it teaches you and how this concept assists you in solving complex problems. Just go with the further section and gain detailed knowledge about the solutions of linear equations, methods to solve the system of linear equations, examples on simultaneous equations, etc.

- Solution of a Linear Equation in Two Variables
- Method of Elimination
- Method of Substitution
- Method of Cross Multiplication

## What is Simultaneous Linear Equation with Example?

Two linear equations in two or three variables determined simultaneously to get a common solution is known as simultaneous linear equations. Simultaneous linear equations in two variables include two strange quantities to depict real-life problems. It aids in setting up a relationship between quantities, prices, speed, time, distance, etc. outcome in a better knowledge of the problems.

The result of the system of simultaneous linear equations is the ordered pair (x, y) that meets both the linear equations.

**For Example,** x + y = 20 and x – y = 10 are two linear equation (simultaneous equations). In case, we take x = 15 and y = 5, then the two equations are met, so we can say the solution of the system of given simultaneous linear equations is (10, 5).

### Important steps to form & solve Simultaneous Linear Equations

Let’s consider any mathematical question to denote and understand the important steps for forming simultaneous equations:

*In a book shop, the cost of 3 books exceeds the cost of 4 pencils by $6. Also, the total cost of 5 books and 7 pencils is $20.*

**Step 1:** Detect the unknown variables; let’s take one of them as x and the other as y.

Here two unknown variables are:

Cost of each book = $x

Cost of each pencil = $y

**Step 2:** Now, find the relation between the unknown quantities.

Cost of 3 books = $3x

Cost of 4 pencils = $4y

Hence, the first condition is 3x-4y = 6

**Step 3:** Communicate the given conditions of the problem in terms of x and y

Now, go for the other condition ie., cost of 5 books = $5x

cost of 7 pencils = $7y

Therefore, the second condition is 5x+7y = 10

Simultaneous equations formed from the given conditions of the problem:

(i) 3x-4y = 6

(ii) 5x+7y = 10

### How To Solve Simultaneous Linear Equations Using Substitution Method?

From this section, you can explore the detailed steps on how to solve a system of linear equations (Simultaneous equations) using the substitution method. They are as follows:

- Firstly, obtain a relation that divides one of the variables by changing the subject of a formula.
- Now, substitute the relation into the other expression(s) to diminish the no. of variables by 1.
- Just go with the same steps till we’re left with a single variable, and solve for it.
- Finally, at this step, we have to substitute the end result in the relations and state the complete solution.

### Steps for Solving Simultaneous Linear Algebraic Equations via Elimination Method?

The process of the Elimination method is repeated until the values of n variables are found. Want to learn the simple steps to solve the system of linear equations using the elimination method? then look at the below lines:

- First, identify two equations that have the same variable.
- Now, perform the multiplication operation on each equation by a number to become their coefficients equal.
- Next, subtract the two equations
- You need to do the same process until you are left with a single variable, and solve for it.
- At last, substitute the resultant value into the original equations and find the values of unknown variables.

### Why Simultaneous Linear Equations Usage?

- To memorize the process of framing Simultaneous Linear Equations from Mathematical Problems.
- By using the method of comparison and method of elimination, we can remember to solve the simultaneous equations.
- For obtaining the ability to find simultaneous equations via the method of substitution and method of cross-multiplication.
- To take the skill to solve mathematical problems framing simultaneous equations.
- To understand the condition for a pair of linear expressions to become simultaneous equations

### Problems on Simultaneous Linear Equations

**Example 1: **

Show that x = 3 and y = 6 is the solution of the system of linear equation x + 2y = 15 and 2x + y =12.

**Solution:**

Given that equation 1 is x + 2y = 4

equation 2 is x + 3y = 8

Now, put x = 3 and y = 6 in equation 1 ie., x + 2y = 15

LHS = x + 2y = 3 + 2×6 = 3 + 12 = 15. which is equal to RHS.

Now, put x = 3 and y = 6 in LHS of 2nd equation ie., 2x + y =12

LHS = 2x + y = 2×3 + 6 = 6 + 6 = 12, which is equal to RHS.

Thus, x = 3 and y = 6 is the solution of the given system of equations.

**Example 2:**

Consider the two following simultaneous linear equations:

x + y = 5 ………… (i)

2x – 3y = 15 ………… (ii)

**Solution:**

The given equations are:

x + y = 5 ………… (i)

2x – 3y = 15 ………… (ii)

From (i) we get y = 5-x

Now, substituting the y value in equation (ii) then we get;

2x – 3(5-x) = 15

2x -15 + 3x = 15

2x + 3x = 30

5x = 30

x = 5

Substitute the solved x value in equation (i), we get;

x + y = 5

5 + y = 5

y = 0

Hence, (5, 0) is the solution of the system of equation x + y = 5 and 2x – 3y = 15.

### FAQs on Framing Simultaneous Equations

1. What are simultaneous equations?

The two or more algebraic equations that share variables e.g. x and y are called simultaneous equations.

2. What is the graph of two linear equations?

The graph of two linear equations is Two straight lines.

3. Where on the graph do we get the solution of a system of linear equations?

At the point of intersection of two lines, we got to know a pair of simultaneous equations is equivalent.