# Simultaneous Equation

In this chapter, we will be discussing the below-mentioned topics in detail:

• Solving Simultaneous Equations using the substitution and elimination method
• Formulate a pair of linear equations in two variables to solve mathematical and real-life problems.

## Graphical Method

1. Draw the graphs of the given equations.
2. Find the coordinates of the point of intersection.
3. Express coordinates as a solution.
i.e. \begin{align*} x &= \text{__________} \end{align*}, \begin{align*} y &= \text{__________} \end{align*}.

## Substitution Method

1. Make x or y the subject of one of the equations.
2. Substitute that equation into the other equation.

## Elimination Method

1. Make sure the coefficients of one variable are the same.
2. Add or subtract the equations to eliminate one variable.

Let’s understand this with the help of some examples:

Question 1:

Solve the simultaneous equations using Substitution Method.

\begin{align} 2x &= 4 + 3y \\ \\ 3 &= x-y \end{align}

Solution:

Using substitution method,

Let us label these equations as Equation (1) and Equation (2),

\begin{align} 2x &= 4 + 3y & \text{.......... (1)} \\ \\ 3 &= x-y & \text{.......... (2)} \end{align}

Method 1:

From equation (2), make $x$ the subject.

\begin{align} 3+y &=x & \text{.......... (3)} \end{align}

Substituting (3) into (1),

\begin{align*} 2(3+y) &= 4+3y \\ \\ 6+2y &= 4+3y \\ \\ 2y-3y &= 4-6 \\ \\ -y &=-2 \\ \\ \therefore \qquad\quad y&=2 \end{align*}

Substituting $y=2$ into Equation (3),

\begin{align*} 3+2 &=x \\ \\ \therefore\qquad x &=5 \end{align*}

Hence, $x=5$, $y=2$.

Method 2:

From equation (2), make y the subject.

\begin{align} y &= x-3 & \text{.......... (4)} \end{align}

Substituting (4) into (1),

\begin{align*} 2x&=4+3(x-3)\\\\ 2x&=4+3x-9\\\\ 2x-3x&=-5\\\\ -x&=-5\\\\ \therefore\qquad\quad x&=5 \end{align*}

Hence, $x=5$

Substituting $x=5$  into Equation (3)

\begin{align*} y&=5-3\\\\ \therefore\quad y&=2 \end{align*}

Hence, $x=5$, $y=2$.

Question 2:

Solve the simultaneous equations.

\begin{align} 3x+2y &=14 \\ \\ 4y &=13-x \end{align}

Solution:

Using substitution method,

Let us label these equations as Equation (1) and Equation (2),

\begin{align} 3x+2y &=14 & \text{.......... (1)} \\ \\ 4y &=13- x & \text{.......... (2)} \end{align}

From equation (2), make $x$ the subject.

\begin{align} x &=13-4y & \text{.......... (3)} \end{align}

Substituting (3) into (1),

\begin{align*} 3(13-4y)+2y &=14 \\ \\ 39-12y+2y &=14 \\ \\ -25&=-10y \end{align*}

Hence, \begin{align} y = 2\frac {1}{2} \end{align}

Substituting\begin{align} y = 2\frac {1}{2} \end{align}into Equation (3),

\begin{align*} x&=13-4(2\frac{1}{2}) \\ \\ &=3 \\ \\ \therefore\quad x &=3 \quad \& \quad y = 2\frac {1}{2} \end{align*}

Question 3:

Solve the simultaneous equations using Elimination Method.

\begin{align} 4x+y &=3 \\ \\ 2x-3y &=12 \end{align}

Solution:

Using elimination method,

Let us label these equations as Equation (1) and Equation (2),

\begin{align} 4x+y &=3 & \text{.......... (1)} \\ \\ 2x-3y &=12 & \text{.......... (2)} \end{align}

From equation (2), we multiply 2 throughout the equation.

\begin{align} 4x-6y &=24 & \text{.......... (3)} \end{align}

Taking \begin{align} \text{(1) – (2)} \end{align},

\begin{align*} (4x+y)-(4x-6y)&=3-24\\\\ 7y&=-21\\\\y&=-3 \end{align*}

Substituting\begin{align} y &=-3 \end{align}into Equation (1),

\begin{align*} 4x+(-3) &=3 \\ \\ 4x &=6 \\ \\ x &= 1\frac {1}{2} \\ \\ \therefore\qquad\qquad x &= 1\frac{1}{2} \quad \& \quad y=-3 \end{align*}

Continue Learning
Algebraic Fractions Direct & Inverse Proportion
Congruence And Similarity Factorising Quadratic Expressions
Further Expansion And Factorisation Quadratic Equations And Graphs
Simultaneous Equation
Primary
Secondary
Book a free product demo
Suitable for primary & secondary
Our Education Consultants will get in touch with you to offer your child a complimentary Strength Analysis.
Book a free product demo
Suitable for primary & secondary
our educational content
Start practising and learning.
No Error
No Error
*By submitting your phone number, we have
your permission to contact you regarding
Let’s get learning!
resources now.
Error
Oops! Something went wrong.
Let’s refresh the page!
Turn your child's weaknesses into strengths
Turn your child's weaknesses into strengths
Trusted by over 220,000 students.

Error
Oops! Something went wrong.
Let’s refresh the page!
Error
Oops! Something went wrong.
Let’s refresh the page!