Simultaneous Equations (Junior Cert Mathematics): Revision Notes

Worked Example: Solving Simultaneous Equations Using Elimination

Let's solve the following simultaneous equations: $\text{Equation A: } \quad 2x - 3y = 7$ $\text{Equation B: } \quad 3x + 2y = 4$

Step 1: Label the Equations

What we do:

We start by labelling the two equations as Equation A and Equation B. This helps us keep track of them as we work through the problem. So we have:

$\text{Equation A: } \quad 2x - 3y = 7$ $\text{Equation B: } \quad 3x + 2y = 4$

Why we do it:

Labelling the equations makes it easier to reference them clearly as we proceed with solving.

Step 2: Multiply the Equations to Make One Variable Cancel Out

What we do:

Our goal is to eliminate one of the variables, either $x$ or $y$, by making their coefficients the same. To do this, we look at the coefficients of the variables in both equations.

• In Equation A, the coefficient of $x$ is $2$, and the coefficient of $y$ is $-3.$
• In Equation B, the coefficient of $x$ is $3$, and the coefficient of $y$ is $2$. To eliminate one variable, we need the coefficients of one of the variables to be the same in both equations. We can choose to eliminate either $x$ or $y$. Let's eliminate $y$ by making their coefficients the same (but with opposite signs so that they cancel out when added).

To eliminate $y$, we need to:

• Multiply Equation A by $2$ (to make the coefficient of $y$ equal to $-6$).
• Multiply Equation B by $3$ (to make the coefficient of $y$ equal to $6$). Multiplying Equation A by 2**:** $2(2x - 3y) = 2(7)$ This gives us: $4x - 6y = 14 \quad \text{(Equation C)}$

Multiplying Equation B by $3$: $3(3x + 2y) = 3(4)$ This gives us: $9x + 6y = 12 \quad \text{(Equation D)}$

Why we do it:

Multiplying the equations by these numbers makes the coefficients of $y$ the same (with opposite signs) in both equations, allowing us to eliminate $y$ by adding the two equations together.

Step 3: Add the Equations to Eliminate One Variable

What we do:

Now that we have made the coefficients of $y$ the same but with opposite signs, we can add the two equations together to eliminate $y$.

Add Equation C and Equation D: $(4x - 6y) + (9x + 6y) = 14 + 12$

When we add them, the $y$ terms cancel out: $4x + 9x - 6y + 6y = 26$

This simplifies to: $13x = 26$

Why we do it:

By adding the equations, the $y$ variable is eliminated, leaving us with a simpler equation that only involves $x$.

Step 4: Solve for the Remaining Variable

What we do:

Now, solve the equation we've just found for $x$.

Divide both sides by $13$ to solve for $x$: $x = \frac{26}{13} = 2$

This gives us the value of $x$.

Why we do it:

This step isolates $x$, giving us its value. With $x$ known, we can find the value of $y.$

Step 5: Substitute Back to Find the Other Variable

What we do:

Now that we know $x = 2$, we can substitute this value back into one of the original equations to find $y$.

Let's substitute $x = 2$ into Equation A: $2x - 3y = 7$ Substituting $x = 2$: $2(2) - 3y = 7$

This simplifies to: $4 - 3y = 7$

Subtract $4$ from both sides: $-3y = 3$

Finally, divide by $-3$: $y = \frac{3}{-3} = -1$

This gives us the value of $y$.

Why we do it:

This step allows us to find the value of $y$ now that we know $x$.

Step 6: Check Your Solutions

What we do:

It's important to check that your solutions work by plugging the values of $x$ and $y$ back into the original equations.

For Equation A: $2x - 3y = 7$ Substituting $x = 2$ and $y = -1$: $2(2) - 3(-1) = 4 + 3 = 7$

For Equation B: $3x + 2y = 4$ Substituting $x = 2$ and $y = -1$: $3(2) + 2(-1) = 6 - 2 = 4$

Both equations are satisfied, so our solutions are correct.

Why we do it:

Checking ensures that our solutions are correct and satisfy both original equations. This is an important step to confirm that we haven't made any errors along the way.