SIMULTANEOUS LINEAR EQUATIONS

Definition

Simultaneous linear equations are two or more equations with the same variables that are solved at the same time to find values that satisfy all equations.

Form

For two variables x and y, the general form is:

$\boldsymbol{a_{1}x + b_{1}y = c_{1}}$

$\boldsymbol{a_{2}x + b_{2}y = c_{2}}$

Where:

$\boldsymbol{a_{1}, b_{1}, a_{2}, b_{2}}$ are coefficients
$\boldsymbol{c_{1}, c_{2}}$ are constants

Example

$\boldsymbol{x + y = 10}$

$\boldsymbol{x - y = 2}$

This is a simultaneous linear equation

Methods of Solving Simultaneous Equations

Common methods of calculating simultaneous linear equations are:

Substitution Method
Elimination Method

In this lesson, we'll focus on these two methods.

Other methods are:

Matrix method
Cramer's Rule/Determinant Rule(under matrix)
Cross-Multiplication Method
Graphical Method

1.Substitution Method

Substitution is solving by defining one variable in terms of the other.

Make one variable the subject
Substitute into the other equation
Solve step by step

Example 1

Solve the following simultaneous equations Using substitution Method

$\boldsymbol{2x + 2y = 8}$

$\boldsymbol{3x + y = 2}$

solution :

(i)Make x the subject of the formula in the first equation.

$\boldsymbol{2x = 8 - 2y}$

$\boldsymbol{x = \frac{8 - 2y}{2}}$

$\boldsymbol{x = (4 - y)}$

(ii)Substitute x in the second equation.

$\boldsymbol{3(4 - y) + y = 2}$

(iii)Solve for y.

$\boldsymbol{12 - 3y + y = 2}$

$\boldsymbol{-3y + y = 2 - 12}$

$\boldsymbol{-2y = -10}$

$\boldsymbol{y = 5}$

(iv)solve for x

In step 1,we found that x = (4 - y)

To find x, we simply substitute the value of y in this equation

$\boldsymbol{x = (4 - 5)}$

$\boldsymbol{x = -1}$

Final Answers:

$\boldsymbol{x = -1}$

$\boldsymbol{y = 5}$

Example 2

Solve the following pair of equations using substitution method :

$\boldsymbol{2x + y = 8}$

$\boldsymbol{3x + y = 11}$

Step 1: Make y the subject of the first equation.

$\boldsymbol{y = 8 - 2x}$

This expresses y in terms of x.

Step 2: Substitute y into the second equation.

$\boldsymbol{3x + (8 - 2x) = 11}$

This replaces y with its equivalent expression.

Step 3: Simplify the equation.

$\boldsymbol{3x + 8 - 2x = 11}$

This removes the brackets.

Step 4: Solve for x.

$\boldsymbol{x + 8 = 11}$

$\boldsymbol{x = 3}$

This gives the value of x.

Step 5: Substitute x into y = 8 - 2x.

$\boldsymbol{y = 8 - 2(3)}$

This helps us find the value of y.

Step 6: Simplify and solve for y.

$\boldsymbol{y = 8 - 6}$

$\boldsymbol{y = 2}$

This gives the value of y.

Final Answers:

$\boldsymbol{x = 3}$

$\boldsymbol{y = 2}$

Example 3

Solve the following pair of equations using substitution meethod :

$\boldsymbol{2x + 3y = 13}$

$\boldsymbol{3x + 2y = 12}$

Step 1: Make y the subject of the first equation.

$\boldsymbol{3y = 13 - 2x}$

$\boldsymbol{y = \frac{13 - 2x}{3}}$

This expresses y in terms of x.

Step 2: Substitute y into the second equation.

$\boldsymbol{3x + 2\left[\frac{13 - 2x}{3}\right] = 12}$

This replaces y with its equivalent expression.

Step 3: Multiply through by 3 to remove the denominator.

$\boldsymbol{9x + 2(13 - 2x) = 36}$

This simplifies the equation.

Step 4: Expand the brackets.

$\boldsymbol{9x + 26 - 4x = 36}$

This removes the brackets.

Step 5: Simplify the equation.

$\boldsymbol{5x + 26 = 36}$

This combines like terms.

Step 6: Solve for x.

$\boldsymbol{5x = 10}$

$\boldsymbol{x = 2}$

This gives the value of x.

Step 7: Substitute x into y = (13 - 2x) / 3.

$\boldsymbol{y = \frac{13 - 4}{3}}$

This helps us find the value of y.

Step 8: Simplify and solve for y.

$\boldsymbol{y = \frac{9}{3}}$

$\boldsymbol{y = 3}$

This gives the value of y.

Final Answers:

$\boldsymbol{x = 2}$

$\boldsymbol{y = 3}$

2.Elimination Method

Elimination is solving by cancelling one variable through the addition or subtraction of the equations.

Make coefficients equal
Add or subtract equations
Eliminate one variable

Example 1

Solve the following simultaneous equations Using Elimination Method

$\boldsymbol{2x + 2y = 8}$

$\boldsymbol{3x + y = 2}$

Solution :

(i)Make x coefficients to be equal by "Cross multiplying the coefficients"

$\boldsymbol{3(2x + 2y = 8)}$

$\boldsymbol{2(3x + y = 2)}$

$\boldsymbol{6x + 6y = 24}$

$\boldsymbol{6x + 2y = 4}$

(ii)Subtract the equations to eliminate the x variable

(iii)After subtracting,you get:

$\boldsymbol{0 + 4y = 20}$

(iv)Solve for y

$\boldsymbol{4y = 20}$

$\boldsymbol{y = 5}$

(v)Solve for x

(vi)Substitute y in any equation containing both the coefficients

(vii)I'll choose the second equation(Both equations will give the correct Answer): $\boldsymbol{3x + y = 2}$

$\boldsymbol{3x + y = 2}$; but $\boldsymbol{y = 5}$

$\boldsymbol{3x + 5 = 2}$

$\boldsymbol{3x = 2 - 5}$

$\boldsymbol{3x = -3}$

$\boldsymbol{x = -1}$

Final Answers:

$\boldsymbol{x = -1}$

$\boldsymbol{y = 5}$

Example 2

Solve the following simultaneous equations Using Elimination Method

$\boldsymbol{2x + y = 8}$

$\boldsymbol{3x + y = 11}$

Step 1: Subtract the first equation from the second equation.

$\boldsymbol{(3x + y) - (2x + y) = 11 - 8}$

This step eliminates the variable y.

Step 2: Simplify the equation.

$\boldsymbol{x = 3}$

This gives the value of x.

Step 3: Substitute x into the first equation.

$\boldsymbol{2(3) + y = 8}$

This step helps us find the value of y.

Step 4: Simplify and solve for y.

$\boldsymbol{6 + y = 8}$

$\boldsymbol{y = 2}$

This gives the value of y.

Final Answers:

$\boldsymbol{x = 3}$

$\boldsymbol{y = 2}$

Example 3

Solve the following simultaneous equations Using Elimination Method

$\boldsymbol{2x + 3y = 13}$

$\boldsymbol{3x + 2y = 12}$

Step 1: Multiply the first equation by 2.

$\boldsymbol{4x + 6y = 26}$

This makes the coefficients of y easier to eliminate.

Step 2: Multiply the second equation by 3.

$\boldsymbol{9x + 6y = 36}$

This makes the coefficients of y equal.

Step 3: Subtract the first new equation from the second new equation.

$\boldsymbol{(9x + 6y) - (4x + 6y) = 36 - 26}$

This step eliminates the variable y.

Step 4: Simplify the equation.

$\boldsymbol{5x = 10}$

This gives a simple linear equation.

Step 5: Solve for x.

$\boldsymbol{x = 2}$

This gives the value of x.

Step 6: Substitute x into the first equation.

$\boldsymbol{2(2) + 3y = 13}$

This step helps us find the value of y.

Step 7: Simplify and solve for y.

$\boldsymbol{4 + 3y = 13}$

$\boldsymbol{3y = 9}$

$\boldsymbol{y = 3}$

This gives the value of y.

Final Answers:

$\boldsymbol{x = 2}$

$\boldsymbol{y = 3}$

Sample Questions

Attempt the following Questions:

(1.)

$\boldsymbol{x + y = 5}$

$\boldsymbol{x - y = 1}$

(2.)

$\boldsymbol{x + y = 10}$

$\boldsymbol{x + 2y = 12}$

(3.)

$\boldsymbol{2x + y = 7}$

$\boldsymbol{x + y = 4}$

(4.)

$\boldsymbol{3x + y = 10}$

$\boldsymbol{x - y = 2}$

(5.)

$\boldsymbol{x + y = 8}$

$\boldsymbol{2x - y = 7}$

Check The Answers Below:

TAKE NOTE :

Understand simultaneous equations: These are equations solved together to find values that satisfy all equations at the same time.
Solutions must satisfy both equations: Always ensure your final answers work in each equation given.
Substitution Method: This method involves making one variable the subject of one equation and substituting it into the other equation.
Choose the easiest variable: In substitution, always pick the variable that is simplest to isolate to reduce errors.
Substitute correctly: After making one variable the subject, replace it carefully in the second equation without changing signs.
Simplify fully: After substitution, simplify the equation completely before solving for the variable.
Elimination Method: This method involves adding or subtracting equations to eliminate one variable.
Make coefficients equal: Before eliminating, multiply equations if necessary so that one variable has equal coefficients.
Eliminate carefully: Add or subtract the equations correctly to remove one variable completely.
Watch negative signs: Be careful when subtracting equations, as sign errors are very common.
Solve step by step: After eliminating one variable, solve for the remaining variable, then substitute back to find the other.
Check your answers: Always substitute your values back into both original equations to confirm correctness.
Avoid unnecessary complexity: Choose the method (substitution or elimination) that makes the solution easiest.
Practice regularly: Mastery comes from solving different types of questions using both methods.

SIMULTANEOUS LINEAR EQUATIONS

Definition

Form

Example

Methods of Solving Simultaneous Equations

1.Substitution Method

Example 1

Example 2

Example 3

2.Elimination Method

Example 1

Example 2

Example 3

Sample Questions

Check The Answers Below:

Answers:

TAKE NOTE :

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