Lesson Explainer: Representing a System of Two Equations in Matrix Form Mathematics

In this explainer, we will learn how to represent a system of two equations as a matrix equation.

Before we begin, it is important that we be confident with matrix multiplication, as this is necessary in the definition of representing a system of equations in terms of matrices. We know how to solve a system of equations using methods such as elimination and substitution. However, in this explainer, we will learn how to represent a system of linear equations as a matrix equation.

Now, before we look at some examples where we will represent systems of two equations in matrix form, let us look at a general form for this process.

How To: Representing a System of Two Linear Equations in Matrix Form

If we have the system of equations 𝑎𝑥+𝑏𝑦=𝑒,𝑐𝑥+𝑑𝑦=𝑓, then to represent this as a matrix equation, we will begin by creating a 2×2 matrix of the coefficients of the variables. We know that this will be a 2×2 matrix as we have two unknowns in each of the two equations.

This gives us the following matrix, which is known as the coefficient matrix: 𝑎𝑏𝑐𝑑.

Next, we complete the matrix equation by stating that this matrix multiplied by the 2×1variables matrix 𝑥𝑦, where 𝑥 and 𝑦 are our unknowns, is equal to the 2×1 matrix 𝑒𝑓, where 𝑒 and 𝑓 are our answer values. This is sometimes also known as the constants matrix.

The full matrix equation is shown below: 𝑎𝑏𝑐𝑑𝑥𝑦=𝑒𝑓.

It is important to emphasize that, in the coefficient matrix, the values take the signs of those in the system of equations.

We can use matrix multiplication to show how this matrix equation works; by using matrix multiplication, the left-hand side of this equation simplifies to give 𝑎𝑏𝑐𝑑𝑥𝑦=𝑎𝑥+𝑏𝑦𝑐𝑥+𝑑𝑦.

Then, this matrix must be equal to our constants matrix: 𝑎𝑥+𝑏𝑦𝑐𝑥+𝑑𝑦=𝑒𝑓.

Finally, these matrices are equal exactly when our original system is solved.

Now, we will look at some examples.

Example 1: Expressing a Pair of Simultaneous Equations as a Matrix Equation

Express the simultaneous equations 3𝑎+2𝑏=13,2𝑎+3𝑏=7 as a matrix equation.

Answer

We recall that if we have a system of two equations in the form 𝑎𝑥+𝑏𝑦=𝑒,𝑐𝑥+𝑑𝑦=𝑓, then this can be represented as a matrix equation in the form 𝑎𝑏𝑐𝑑𝑥𝑦=𝑒𝑓, where 𝑎𝑏𝑐𝑑 is the matrix of coefficients, with the first column being the 𝑥-coefficients and the second being the 𝑦-coefficients, 𝑥𝑦 is the matrix of the variables, and 𝑒𝑓 is the matrix of the answers to each of the equations.

Therefore, the pair of equations in this question can be represented as the matrix equation 3223𝑎𝑏=137.

In the next example, we will solve a problem that includes fractional coefficients and answers to demonstrate that the method remains unchanged.

Example 2: Expressing a Pair of Simultaneous Equations as a Matrix Equation

Express the simultaneous equations 13𝑥23𝑦=53,34𝑦+14𝑥=74 as a matrix equation.

Answer

Firstly, from inspection, we can see that the second equation is in the form 𝑑𝑦+𝑐𝑥=𝑓.

Therefore, we need to consider this when representing this in our matrix equation, as it is a common mistake to have the elements in the wrong order in the coefficient matrix. To avoid this, we should rewrite the system of equations as 13𝑥23𝑦=53,14𝑥+34𝑦=74.

Next, we recall that if we have a system of two equations in the form 𝑎𝑥+𝑏𝑦=𝑒,𝑐𝑥+𝑑𝑦=𝑓, then this can be represented as a matrix equation in the form 𝑎𝑏𝑐𝑑𝑥𝑦=𝑒𝑓, remembering that the elements in the coefficient matrix take the signs of the coefficients of the variables in the original equations.

Therefore, bearing this in mind, we can represent our system of equations as the matrix equation 13231434𝑥𝑦=5374.

In the first two questions, we have represented systems of equations as a matrix equation. In our next two examples, we will write down the set of simultaneous equations that can be represented using a given matrix equation.

Example 3: Identifying a Pair of Simultaneous Equations from a Matrix Equation

Write down the set of simultaneous equations that could be solved using the matrix equation 3324𝑎𝑏=1012.

Answer

We can multiply the two matrices on the left-hand side of the equation to get 3𝑎+3𝑏2𝑎+4𝑏=1012.

Since these matrices must be equal, their entries must be equal; therefore, the set of simultaneous equations that can be solved is 3𝑎+3𝑏=10,2𝑎+4𝑏=12.

In our next example, once again we will be finding a set of equations that can be represented by a given matrix equation; however, this time we will also be considering coefficients with different signs.

Example 4: Identifying a Pair of Simultaneous Equations from a Matrix Equation

Write down the set of simultaneous equations that could be solved using the matrix equation 11394𝑥𝑦=813.

Answer

In this problem, we can multiply the two matrices on the left-hand side of the equation to get 11𝑥3𝑦9𝑥+4𝑦=813.

Since these matrices must be equal, their entries must be equal; therefore, the set of simultaneous equations that can be solved is 11𝑥3𝑦=8,9𝑥+4𝑦=13.

Now, we will explore an example of representing a pair of simultaneous equations as a matrix equation that requires some algebraic manipulation.

Example 5: Expressing a Pair of Simultaneous Equations as a Matrix Equation

Express the simultaneous equations 3𝑥24=8𝑦,𝑥=3𝑦 as a matrix equation.

Answer

First, we will rearrange the equations into the form 𝑎𝑥+𝑏𝑦=𝑒,𝑐𝑥+𝑑𝑦=𝑓, as this will enable us to effectively represent our system of equations as a matrix equation.

We will start with the first equation, 3𝑥24=8𝑦.

We can add 8𝑦 and add 24 to each side of the equation to give us 3𝑥+8𝑦=24.

Next, we rearrange the second equation by adding 𝑦 to each side. This gives us 𝑥+𝑦=3.

Now, we have the pair of equations 3𝑥+8𝑦=24,𝑥+𝑦=3.

We can recall that a system of two equations in the form 𝑎𝑥+𝑏𝑦=𝑒,𝑐𝑥+𝑑𝑦=𝑓 can be represented as a matrix equation in the form 𝑎𝑏𝑐𝑑𝑥𝑦=𝑒𝑓. where we recall that the terms 𝑥 and 𝑦 have coefficients of 1.

Therefore, the pair of equations in this question can be represented as the matrix equation 3811𝑥𝑦=243.

In our final example, we will investigate a problem that includes a zero element in the coefficient matrix.

Example 6: Identifying a Pair of Simultaneous Equations from a Matrix Equation That Includes a Zero Element

Which of the following systems of equations can be represented by the matrix form 0234𝑥𝑦=56?

  1. 2𝑦=5,
    3𝑥4𝑦=6
  2. 2𝑦=5,
    3𝑥4𝑦=6
  3. 2𝑦=5,
    3𝑥4𝑦=6
  4. 2𝑦=5,
    3𝑥4𝑦=6
  5. 2𝑦=5,
    2𝑦=5

Answer

In this problem, we can multiply the two matrices on the left-hand side of the equation to get 0𝑥2𝑦3𝑥4𝑦=56.

Since these matrices must be equal, their entries must be equal; therefore, the set of simultaneous equations that can be represented is 0𝑥2𝑦=5,3𝑥4𝑦=6.

However, we do not include variables with a zero coefficient; therefore, we can rewrite the equations as 2𝑦=5,3𝑥4𝑦=6, which is option A.

We will finish by recapping the key points from this explainer.

Key Points

  • A system of two equations in the form 𝑎𝑥+𝑏𝑦=𝑒,𝑐𝑥+𝑑𝑦=𝑓 can be represented as a matrix equation in the form 𝑎𝑏𝑐𝑑𝑥𝑦=𝑒𝑓, where 𝑎𝑏𝑐𝑑 is known as the coefficient matrix and 𝑒𝑓 as the constants matrix.
  • The elements in the coefficient matrix take the signs of the coefficients of the variables in the original equations.

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