Linear Equations in Two Variables

Vikas PatelX-B

Roll.no -35Maths Project Work

Kendriya Vidyalaya, Ujjain

Lesson 3

A system of equations is a collection of two or more equations, each containing one or more

variables.

A solution of a system of equations consists of values for the variables that reduce each equation

of the system to a true statement.

When a system of equations has at least one solution, it is said to be consistent; otherwise it is

called inconsistent.

To solve a system of equations means to find all solutions of the system.

If each equation in a system of equations is linear, then we have a system of

linear equations.

If the graph of the lines in a system of two linear equations in two variables intersect, then the system of equations has one solution, given by the point of

intersection. The system is consistent and the equations are independent.

Solution

y

x

If the graph of the lines in a system of two linear equations in two variables are parallel, then the

system of equations has no solution, because the lines never intersect. The system is inconsistent.

x

y

If the graph of the lines in a system of two linear equations in two variables are coincident, then the system of equations has infinitely many solutions,

represented by the totality of points on the line. The system is consistent and dependent.

x

y

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 7

Equations of the form ax + by = c are called linear equations in two variables.

The point (0,4) is the y-intercept.

The point (6,0) is the x-intercept.

x

y

2-2

This is the graph of the equation 2x + 3y = 12.

(0,4)

(6,0)

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 8

y

x2-2

The slope of a line is a number, m, which measures its steepness.

m = 0

m = 2m is undefined

m =1

2

m = -1

4

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 9

Two lines are parallel if they have the same slope.

If the lines have slopes m1 and m2, then the lines are parallel whenever m1 = m2.

Example: The lines y = 2x – 3 and y = 2x + 4 have slopes m1 = 2 and m2 = 2.

The lines are parallel.

x

y

y = 2x + 4

(0, 4)

y = 2x – 3

(0, -3)

Two Algebraic Methods for Solving a System

1. Method of substitution

2. Method of elimination

STEP 1: Solve for x in (2)

Use Method of Substitution to solve:

(1)

(2)

add x and subtract 2on both sides

STEP 2: Substitute for x in (1)

STEP 3: Solve for y

STEP 4: Substitute y = -11/5 into result in Step1.

Solution:

STEP 5: Verify solution

in

Rules for Obtaining an Equivalent System of Equations

(Elimination Method)

1. Interchange any two equations of the system.

2. Multiply (or divide) each side of an equation by the same nonzero constant.

3. Replace any equation in the system by the sum (or difference) of that equation and any other equation in the system.

Multiply (2) by 2

Replace (2) by the sum of (1) and (2)

Equation (2) has no solution. System is inconsistent.

Use Method of Eliminationto solve:

(1)

(2)

Multiply (2) by 2

Replace (2) by the sum of (1) and (2)

The original system is equivalent to a system containing one equation. The equations are dependent.

Use Method of Eliminationto solve:

(1)

(2)

This means any values x and y, for which 2x -y =4 represent a solution of the system.

Thus there is infinitely many solutions and they can be written as

or equivalently

Solve by Graphing? Substitution?

• This is practical for two-variable systems, but very impractical once problem gets larger:

• Solve the following system:

3x + 5y = -9

x+ 4y = -10

Solve by Substitution3x + 5y = -9

x+ 4y = -10

Substitution: Solve equation 1 for x, replace for x in second equation.

3x =- 5y -9 , x=(-5/3)y-3

(-5/3)y-3+ 4y = -10

(-5/3)y+(12/3)y=-7

(7/3)y=-7

Y=-3

Plug back into either equation to solve for x.

3x+5(-3)=-93x=6x=2… OR x+4(-3)=-10x=2

Solution? (2,-3)

Method of Substitution

• The method of substitution is an algebraic one. It works well when the coefficients of x or y are either 1 or -1. For example, let’s solve the system

2x + 3 = y

x + 2y = -4

using the method of substitution.

Method of Substitution(continued)

The steps for this method are as follows:

1) Solve one of the equations for either x or y.

2) Substitute that result into the other equation to obtain an equation in a single variable (either x or y).

3) Solve the equation for that variable.

4) Substitute this value into any convenient equation to obtain the value of the remaining variable.

Summary• A linear equation in two unknowns is an equation that can be written in the form

• ax + by = c with a, b and c being real numbers, and with a and b not both zero.

• A solution to a linear equation in the two unknowns x and y consists of a pair of numbers: a value for x and a value for y that satisfy the equation. By the same token, a solution to a system of two or more linear equations in x and y is a solution that satisfies all of the equations in the system.

• We can solve such a system of equations either graphically, by drawing the graphs and finding where they intersect, or algebraically, by combining the equations in order to eliminate all but one variable and then solving for that variable.

• A system of two linear equations in two unknowns has either:

• (1) A single (or unique) solution. This happens when the lines corresponding to the two equations are not parallel, so that they intersect at a single point.

• (2) No solution. This happens when the two lines are parallel and different.

• (3) An infinite number of solutions. This occurs when the two equations represent the same straight line. In this case, we represent the solutions by choosing one variable arbitrarily, and solving for the other.

23