# Linear Equations in Two Variables (Definition and Solutions) (2023)

Linear equations in two variables, explain the geometry of lines or the graph of two lines, plotted to solve the given equations. As we already know, the linear equation represents a straight line. The plotting of these graphs will help us to solve the equations, which consist of unknown variables. Previously we have learned to solve linear equations in one variable, here we will find the solutions for the equations having two variables.

• Definition
• Solution
• Example
• Unique Solution
• No Solution
• System of Linear Equation in Two Variables
• Problems and Solutions
• Word Problems
• FAQs

## Definition

An equation is said to be linear equation in two variables if it is written in the form of ax + by + c=0, where a, b & c are real numbers and the coefficients of x and y, i.e a and b respectively, are not equal to zero.

For example, 10x+4y = 3 and -x+5y = 2 are linear equations in two variables.

The solution for such an equation is a pair of values, one for x and one for y which further makes the two sides of an equation equal.

## Solution of Linear Equations in Two Variables

The solution of linear equations in two variables, ax+by = c, is a particular point in the graph, such that when x-coordinate is multiplied by a and y-coordinate is multiplied by b, then the sum of these two values will be equal to c.

Basically, for linear equation in two variables, there are infinitely many solutions.

## Example

In order to find the solution of Linear equation in 2 variables, two equations should be known to us.

Consider for Example:

5x + 3y = 30

The above equation has two variables namely x and y.

Graphically this equation can be represented by substituting the variables to zero.

The value of x when y=0 is

5x + 3(0) = 30

⇒x=6

and the value of y when x = 0 is,

5 (0) + 3y = 30

(Video) Linear Equations in 2 Variables - Review

⇒ y = 10

It is now understood that to solve linear equation in two variables, the two equations have to be known and then the substitution method can be followed. Let’s understand this with a few example questions.

### Unique Solution

For the given linear equations in two variables, the solution will be unique for both the equations, if and only if they intersect at a single point.

The condition to get the unique solution for the given linear equations is, the slope of the line formed by the two equations, respectively, should not be equal.

Consider, m1 and m2 are two slopes of equations of two lines in two variables. So, if the equations have a unique solution, then:

m1 ≠ m2

### No Solution

If the two linear equations have equal slope value, then the equations will have no solutions.

m1 = m2

This is because the lines are parallel to each other and do not intersect.

## System of Linear Equations in Two Variables

Instead of finding the solution for a single linear equation in two variables, we can take two sets of linear equations, both having two variables in them and find the solutions. So, basically the system of linear equations is defined when there is more than one linear equation.

For example, a+b = 15 and a-b = 5, are the system of linear equations in two variables. Because, the point a = 10 and b = 5 is the solution for both equations, such as:

a+b=10 + 5 = 15

a-b=10-5 = 5

Hence, proved point (10,5) is solution for both a+b=15 and a-b=5.

### Problems and Solutions

Question: Find the value of variables which satisfies the following equation:

2x + 5y = 20 and 3x+6y =12.

Solution:

Using the method of substitution to solve the pair of linear equation, we have:

(Video) Linear Equations in two Variables

2x + 5y = 20…………………….(i)

3x+6y =12……………………..(ii)

Multiplying equation (i) by 3 and (ii) by 2, we have:

6x + 15y = 60…………………….(iii)

6x+12y = 24……………………..(iv)

Subtracting equation (iv) from (iii)

3y = 36

⇒y=12

Substituting the value of y in any of the equation (i) or (ii), we have

2x + 5(12) = 20

⇒ x = −20

Therefore, x=-20 and y =12 is the point where the given equations intersect.

Now, it is important to know the situational examples which are also known as word problems from linear equations in 2 variables.

## Word Problems

Question 1:A boat running downstream covers a distance of 20 km in 2 hours while for covering the same distance upstream, it takes 5 hours. What is the speed of the boat in still water?

Solution:

These types of questions are the real-time examples of linear equations in two variables.

In water, the direction along the stream is called downstream. And, the direction against the stream is called upstream.

Let us consider the speed of a boat is u km/h and the speed of the stream is v km/h, then:

(Video) The Definition of a Linear Equation in Two Variables

Speed Downstream = (u + v) km/h

Speed Upstream = (u – v) km/h

We know that, Speed = Distance/Time

So, the speed of boat when running downstream =(20⁄2)km/h=10km/h

The speed of boat when running upstream = (20⁄5)km/h= 4 km/h

From above, u + v = 10…….(1)

u – v = 4 ………. (2)

Adding equation 1 and 2, we get: 2u = 1

u = 7 km/h

Also, v = 3 km/h

Therefore, the speed of the boat in still water = u = 7 km/h

Question 2: A boat running upstream takes 6 hours 30 minutes to cover a certain distance, while it takes 3 hours to cover the same distance running downstream. What is the ratio between the speed of the boat and speed of the water current, respectively?

Solution: If the speed downstream isakm/hrand the speed upstream isbkm/hr, then

Speed in still water = (a + b)/2 km/h

Rate of stream = ½ (a − b) kmph

Let the Boat’s rate upstream bexkmph and that downstream beykmph.

Then, distance covered upstream in 6 hrs 30 min = Distance covered downstream in 3 hrs.

⇒ x × 6.5 hrs =y × 3hrs

⇒ 13/2x=3y

(Video) Two-variable linear equations and their graphs | Algebra I | Khan Academy

⇒ y= 13x/6

$$\begin{array}{l}The\ required\ ratio\ is\ \frac{y + x}{2}~ :~ \frac{y – x}{2}\end{array}$$

$$\begin{array}{l}\Rightarrow~\frac{\frac{13x}{6}~+~x}{2}~:~\frac{\frac{13x}{6}~-~x}{2}\end{array}$$

$$\begin{array}{l}\Rightarrow~\frac{\frac{19x}{6}}{2}~:~\frac{\frac{7x}{6}}{2}\end{array}$$

= 19:7

## Frequently Asked Questions – FAQs

### How to solve linear equation in two variables?

For a system of linear equations in two variables, we can find the solutions by the elimination method.

### How many solutions are there for linear equations in two variables?

For linear equations in two variables, there are infinitely many solutions.

### What is the two-variable equation?

A linear equation in two variables is an equation which has two different solutions.

### What are the coefficients of the equation 3x-6y = -13?

The coefficient of x is 3 and the coefficient of y is -6.

### What is the constant of the equation 3x-6y=-13?

The constant of the equation 3x-6y=-13 is -13.

## FAQs

### What is the definition of the linear equations in two variables? ›

Definition: A linear equation in two variables (x and y) is any equation that can be written in the form ax + by = c with at least one of a and b non-zero. This form is called the General Form of the equation. Not surprisingly, the graph of a linear equation is a straight line.

What are linear equations simple definition? ›

A linear equation is an algebraic equation of the form y=mx+b. involving only a constant and a first-order (linear) term, where m is the slope and b is the y-intercept. Occasionally, the above is called a "linear equation of two variables," where y and x are the variables.

Does a linear equation in two variables have a solution? ›

Further, a linear equation in two variables has infinitely many solutions. The graph of every linear equation in two variables is a straight line and every point on the graph (straight line) represents a solution of the linear equation.

How do you solve a linear equation with two equations? ›

How do I solve systems of linear equations by substitution?
1. Isolate one of the two variables in one of the equations.
2. Substitute the expression that is equal to the isolated variable from Step 1 into the other equation. ...
3. Solve the linear equation for the remaining variable.

How do you solve a linear equation definition? ›

To solve linear equations, find the value of the variable that makes the equation true. Use the inverse of the number that multiplies the variable, and multiply or divide both sides by it. Simplify the result to get the variable value. Check your answer by plugging it back into the equation.

What is the solution of an equation in two variables? ›

In general, a solution of a system in two variables is an ordered pair that makes BOTH equations true. In other words, it is where the two graphs intersect, what they have in common. So if an ordered pair is a solution to one equation, but not the other, then it is NOT a solution to the system.

How do you solve linear equations step by step? ›

1. Step 1: Simplify each side, if needed.
2. Step 2: Use Add./Sub. Properties to move the variable term to one side and all other terms to the other side.
3. Step 3: Use Mult./Div. ...
5. I find this is the quickest and easiest way to approach linear equations.
6. Example 6: Solve for the variable.
Jul 1, 2011

How do you know if a linear equation is a solution? ›

A system of linear equations has one solution when the graphs intersect at a point.

How do you know if an equation has one or two solutions? ›

You can tell that an equation has one solution if you solve the equation and get a variable equal to a number.

How many solutions does a 2 variable equation have? ›

A linear equation in two variables has only 1 solution.

### What are the 3 steps in solving linear equations in two variables? ›

There are three ways to solve systems of linear equations in two variables:
1. graphing.
2. substitution method.
3. elimination method.
Jul 10, 2011

What are the 4 methods of solving linear equations in two variables? ›

Answer: The methods used for solving linear equations in two variables are the graphing method, substitution method, elimination method and matrix method. Let's understand these methods in detail.

What is the best definition for equation? ›

In its simplest form in algebra, the definition of an equation is a mathematical statement that shows that two mathematical expressions are equal. For instance, 3x + 5 = 14 is an equation, in which 3x + 5 and 14 are two expressions separated by an 'equal' sign.

How do you solve linear variables? ›

Solving Linear Equations in One Variable
1. Step 1: Using LCM, clear the fractions if any.
2. Step 2: Simplify both sides of the equation.
3. Step 3: Isolate the variable.

What are the solutions of an equation? ›

A solution of an equation is any value of the variable that satisfies the equality, that is, it makes the Left Hand Side (LHS) and the Right Hand Side (RHS) of the equation the same value. To solve an equation is to find the solution(s) for that equation.

What are the 3 rules to solving an equation? ›

Section 3: A General Rule for Solving Equations
• Simplify each side of the equation by removing parentheses and combining like terms.
• Use addition or subtraction to isolate the variable term on one side of the equation.
• Use multiplication or division to solve for the variable.

How do you simplify and solve linear equations? ›

The steps for solving linear equations are:
1. Simplify both sides of the equation and combine all same-side like terms.
2. Combine opposite-side like terms to obtain the variable term on one side of the equal sign and the constant term on the other.
3. Divide or multiply as needed to isolate the variable.
Oct 6, 2021

What is a solution of a linear line? ›

The solution of a linear system is the ordered pair that is a solution to all equations in the system. One way of solving a linear system is by graphing. The solution to the system will then be in the point in which the two equations intersect.

What linear equations have no solution? ›

A system of linear equations that has no solution is called an inconsistent pair of linear equations. When we consider a system of linear equations, we can compare the coefficients of the equations and find whether it is a system of equations with no solution.

What does no solutions look like? ›

The last type of equation is known as a contradiction, which is also known as a No Solution Equation. This type of equation is never true, no matter what we replace the variable with. As an example, consider 3x + 5 = 3x - 5. This equation has no solution.

### How do you know if a solution is inconsistent or dependent? ›

A system of equations is consistent if it has at least one solution. A system is inconsistent if it has no solution. In a system of two equations in two variables, the equations are dependent if one equation is a multiple of the other. Dependent systems have an infinite number of solutions – every point is a solution.

What is the definition of solution in math? ›

Mathematics A value or values which, when substituted for a variable in an equation, make the equation true. For example, the solutions to the equation x2 = 4 are 2 and -2.

How do you identify many solutions? ›

If solving an equation yields a statement that is true for a single value for the variable, like x = 3, then the equation has one solution. If solving an equation yields a statement that is always true, like 3 = 3, then the equation has infinitely many solutions.

What are the 5 examples of linear equation? ›

Some of the examples of linear equations are 2x – 3 = 0, 2y = 8, m + 1 = 0, x/2 = 3, x + y = 2, 3x – y + z = 3.

What are the 3 different types of solutions a 2 variable system can have? ›

There are three types of systems of linear equations in two variables, and three types of solutions.
• An independent system has exactly one solution pair (x,y). The point where the two lines intersect is the only solution.
• An inconsistent system has no solution. ...
• A dependent system has infinitely many solutions.
May 1, 2022

What is a linear function for dummies? ›

Linear functions are those whose graph is a straight line. A linear function has one independent variable and one dependent variable. The independent variable is x and the dependent variable is y. a is the constant term or the y intercept.

What is linear equation for Grade 7? ›

Any equation is written in the form ax + b = 0 here, 'a' and 'b' are real numbers, and 'x' is a variable, called Linear Equation in one Variable. These equations have the variable of order one only. For example, 5x + 13 = 16 is a linear equation with a single variable.

How do you prove an equation is linear? ›

Any equation in the format y=n(n stands for a number) or x=n will be linear.

What are the 3 types of linear equations? ›

There are three major forms of linear equations: point-slope form, standard form, and slope-intercept form.

How do you explain a linear equation to a child? ›

Linear Equations Represent Lines

To make a line you need two points. Then you can draw a line through those two points. The x and y variables in the linear equation represent the x and y coordinates on a graph. If you plug in a number for x, you can calculate the corresponding number for y.

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