chevron icon chevron icon chevron icon

Simultaneous Equation

Simultaneous Equations are a set of equations with multiple variables that are solved together to find a common solution. They're often used when two or more relationships between variables need to be analyzed simultaneously. By solving the equations together, you find values for the variables that satisfy all the equations simultaneously. This can be done using methods like Graphical MethodSubstitution Method and Elimination Method.

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.
     

Solving Simultaneous Equations using Graphical Method, Substitution Method, Elimination Method
 

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.
  3. Solve and present your answers.

Elimination Method

  1. Make sure the coefficients of one variable are the same.
  2. Add or subtract the equations to eliminate one variable.
  3. Solve and present your answers.

 

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

 

Resources - Academic Topics
icon expand icon collapse Primary
icon expand icon collapse Secondary
icon expand icon collapse
Book a free product demo
Suitable for primary & secondary
select dropdown icon
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
Claim your free demo today!
Claim your free demo today!
Arrow Down Arrow Down
Arrow Down Arrow Down
*By submitting your phone number, we have your permission to contact you regarding Geniebook. See our Privacy Policy.
Geniebook CTA Illustration Geniebook CTA Illustration
Turn your child's weaknesses into strengths
Geniebook CTA Illustration Geniebook CTA Illustration
Geniebook CTA Illustration
Turn your child's weaknesses into strengths
Get a free diagnostic report of your child’s strengths & weaknesses!
Arrow Down Arrow Down
Arrow Down Arrow Down
Error
Oops! Something went wrong.
Let’s refresh the page!
Error
Oops! Something went wrong.
Let’s refresh the page!
We got your request!
A consultant will be contacting you in the next few days to schedule a demo!
*By submitting your phone number, we have your permission to contact you regarding Geniebook. See our Privacy Policy.
1 in 2 Geniebook students scored AL 1 to AL 3 for PSLE
Trusted by over 220,000 students.
Trusted by over 220,000 students.
Arrow Down Arrow Down
Arrow Down Arrow Down
Error
Oops! Something went wrong.
Let’s refresh the page!
Error
Oops! Something went wrong.
Let’s refresh the page!
We got your request!
A consultant will be contacting you in the next few days to schedule a demo!
*By submitting your phone number, we have your permission to contact you regarding Geniebook. See our Privacy Policy.
media logo
Geniebook CTA Illustration
Geniebook CTA Illustration
Geniebook CTA Illustration
Geniebook CTA Illustration Geniebook CTA Illustration
icon close
Default Wrong Input
Get instant access to
our educational content
Start practising and learning.
No Error
arrow down arrow down
No Error
*By submitting your phone number, we have
your permission to contact you regarding
Geniebook. See our Privacy Policy.
Success
Let’s get learning!
Download our educational
resources now.
icon close
Error
Error
Oops! Something went wrong.
Let’s refresh the page!