Example 1: Solving Simultaneous Equations Let's solve the following simultaneous equations:

Step-by-Step Solution:

1. Align the Equations:

• Make sure both equations are written in the same form. Here, both are in the form $ax+by=c$.

2. Write One Equation Under the Other:

• Place the second equation directly below the first one:

3. Choose the Key Letter:

• You need to decide which variable (either $x$ or $y$) to eliminate.
• Notice that the coefficients of $y$ are already the same (both are $+1$), so we'll choose $y$ as our Key Letter to eliminate.

4. Subtract the Equations:

• Since the coefficients of $y$ are the same and have the same sign, subtract the second equation from the first:

• This simplifies to:

5. Solve for $x$:

• Now that y$$ is eliminated, solve for $x$:

6. Substitute $x=5$ Back into One of the Original Equations:

• Use either of the original equations to find the value of y$$. Let's use Equation $2$:

• Subtract $5$ from both sides:

7. Final Answer:

• The solution is $x=5$ and $y=4$.

8. Check the Solution:

• Substitute $x=5$ and $y=4$ back into the original equations to ensure they satisfy both

Example 2: Solving Simultaneous Equations Let's solve the following simultaneous equations:

Step-by-Step Solution:

1. Align the Equations:

• Make sure both equations are written in the same form, so you can easily compare and manipulate them.

2. Write One Equation Under the Other:

• Place the second equation directly below the first one:

3. Choose the Key Letter:

• Decide which variable (either $x$ or $y$) to eliminate.
• Notice that the coefficients of $y$ are already the same in magnitude (both are $2$), but with different signs. This makes it easier to eliminate $y$ by adding the equations.

4. Add the Equations:

• Since the coefficients of $y$ are the same but with opposite signs, add the two equations to eliminate $y$:

• This simplifies to:

5. Solve for $x$:

• Now that $y$ is eliminated, solve for $x$:

6. Substitute $x=3$ Back into One of the Original Equations:

• Use either of the original equations to find the value of $y$. Let's use Equation $2$:

• Simplify and solve for $y$:

7. Final Answer:

• The solution is $x=3$ and $y=3$.

8. Check the Solution:

• Substitute $x=3$ and $y=3$ back into the original equations to ensure they satisfy both:

Example 3: Solving Simultaneous Equations Let's solve the following simultaneous equations:

Step-by-Step Solution:

1. Align the Equations:

• First, ensure that both equations are written in a comparable form, so you can easily manipulate them.

2. Write One Equation Under the Other:

• Place the second equation directly below the first one:

3. Adjust the Equations for Elimination:

• We need to make the coefficients of either $x$ or $y$ the same so that we can eliminate one of the variables.
• Multiply Equation $1$ by $3$ and Equation $2$ by $2$ to make the coefficients of $x$ equal:

4. Eliminate One Variable:

• Subtract Equation $3$ from Equation $4$ to eliminate $x$:

• This simplifies to:

5. Solve for the Remaining Variable:

• Now that we have $y=15$, substitute this value back into one of the original equations (Equation $1$) to find $x$:

• Simplify and solve for $x$:

6. Final Answer:

• The solution is $x=−19$ and $y=15$.

7. Check the Solution:

• Substitute $x=−19$ and $y=15$ back into the original equations to ensure they satisfy both:

Example 4: Solving Simultaneous Equations Given the simultaneous equations:

Step-by-Step Solution:

Step 1: Simplify and align the equations

First, let's rewrite the second equation in a form similar to the first:

So now we have:

Step 2: Make the coefficients of one of the variables the same

To eliminate one of the variables, we need the coefficients of either $x$ or $y$ to be the same. We can do this by multiplying the first equation by $2$:

This gives us:

Now we have:

Step 3: Add the equations to eliminate $y$

Since the coefficients of $y$ in Equation $2$ and Equation $3$ are the same but with opposite signs, we can add these two equations to eliminate $y$:

Simplifying:

Step 4: Substitute $x=−2x$ into one of the original equations to find $y$

Let's use Equation $2$ for substitution:

Simplifying:

Final Answer: The solution to the system of equations is:

$x=−2 \quad and\quad y=3$

Step 5: Check the solution

Finally, substitute $x=−2$ and $y=3$ back into the original equations to ensure they satisfy both:

• For Equation $1$:

• For Equation $2$:

Both equations are satisfied, so the solution is correct!

Worked Example

Step-by-Step Solution:

8. Isolate $y$ in the linear equation:

This is already isolated, so we can use this directly.

9. Substitute $y=2x+3$ into the quadratic equation:

10. Re-arrange the equation to make it equal to zero:

This is our quadratic equation to solve.

11. Factorise the quadratic equation:

So, $x=3$ or $x=−1$.

12. Find $y$ for each $x$ value using the linear equation $y=2x+3$:

• For $x=3$:

• For $x=−1$:

Final Answers:

[$$ (x,y)=(3,9)\quad or\quad (x,y)=(−1,1)