Example: Finding the Intersection by Solving Simultaneous Equations

Equations of the lines:

• Line 1: $2x - y - 2 = 0$ ($Equation A$)
• Line 2: $x - 2y + 2 = 0$ ($Equation B$)

Step 1: Label the equations.

We label the first equation as $Equation A$ and the second as $Equation B$ to make it easier to follow.

Step 2: Make the coefficients of $y$ the same.

The goal is to eliminate one of the variables so that we can solve for the other. In this case, we'll eliminate $y$.

To do this, we need to make the coefficients (the numbers in front of $y$) in both equations the same. This will allow us to eliminate $y$ by adding or subtracting the equations.

• In Equation A, the coefficient of $y$ is $-1$ (since $2x - y - 2 = 0$) is the same as $(2x - 1y - 2 = 0)$.

• In Equation B, the coefficient of $y$ is $-2$. To make the coefficients the same, we can multiply Equation B by $2$. This will make the coefficient of $y$ in Equation B equal to $-4$, which will allow us to eliminate $y$ when we add the equations.

• Multiply Equation $B$ by $2$: $2(x - 2y + 2 = 0) \quad \Rightarrow \quad 2x - 4y + 4 = 0 \quad \text{(Equation C)}$

Now, Equation $A$ and Equation $C$ are:

• Equation $A$: $2x - y - 2 = 0$
• Equation $C$: $2x - 4y + 4 = 0$

Step 3: Subtract the equations to eliminate $y$.

Now that the coefficients of $y$ are different enough, we can subtract Equation C from Equation A. This will eliminate the $y$ terms because they will cancel each other out.

• Subtract Equation C from Equation A: $(2x - y - 2) - (2x - 4y + 4) = 0 - 0$

Simplifying this gives:

$3y - 6 = 0$

Now, $y$ is the only variable left, which allows us to solve for $y$.

Step 4: Solve for $y$.

To solve for $y$, we need to isolate $y$ on one side of the equation.

• First, add $6$ to both sides of the equation: $3y = 6$

• Then, divide both sides by $3$: $y = 2$

Now we know that $y = 2$.

Step 5: Substitute $y = 2$ back into one of the original equations to find $x$.

We've found the value of $y$, but we still need to find the value of $x$. To do this, we substitute $y = 2$ back into one of the original equations. This is called back-substituting. The reason we do this is to find the value of the other variable $(x)$ after we have solved for $(y)$.

Let's use Equation B for this:

$x - 2(2) + 2 = 0$

Simplify this:

$x - 4 + 2 = 0$

Combine like terms:

$x - 2 = 0$

Finally, add $2$ to both sides:

$x = 2$

Now we know that $x = 2$.

Step 6: Write the solution as a coordinate pair.

The values $x = 2$ and $y = 2$ give us the point where the two lines intersect. We can write this as a coordinate pair $(2, 2)$.

So, the point of intersection is $(2, 2)$.

This is the point where the two lines meet on the Cartesian Plane.

Key Tips for Success

• Label Everything Clearly: Use labels like $A$, $B$, and $C$ for your equations to keep track of your work.
• Check Your Work: After finding the values of $(x)$ and $(y)$, substitute them back into the original equations to ensure they satisfy both equations.
• Understand the Connection: Remember that solving simultaneous equations in algebra is directly related to finding the point of intersection in co-ordinate geometry. By practicing these steps, you'll become more confident in solving intersection problems and connecting algebra with co-ordinate geometry!