What is the solution to the system of equations y = x² - 5x + 4 and y = x² - 9x + 18?

Solving the System of Equations

We start with the two equations:

Since both equations are equal to y, we can set them equal to each other:

x² – 5x + 4 = x² – 9x + 18

Next, we can eliminate x² from both sides:

-5x + 4 = -9x + 18

Now, let’s isolate terms involving x. We can start by adding 9x to both sides:

4 = -5x + 9x + 18

This simplifies to:

4 = 4x + 18

Next, we subtract 18 from both sides:

4 – 18 = 4x

This gives us:

-14 = 4x

Now, divide both sides by 4:

x = -rac{14}{4} = -rac{7}{2}

Now that we have the value of x, we can substitute it back into either of the original equations to find y. We will use Equation 1:

y = (-rac{7}{2})² – 5(-rac{7}{2}) + 4

Calculating step by step:

y = rac{49}{4} + rac{35}{2} + 4
Converting 4 to a fraction: y = rac{49}{4} + rac{35}{2} + rac{16}{4}
Now, convert rac{35}{2} to the equivalent fraction: rac{35}{2} = rac{70}{4}
Now, we have: y = rac{49}{4} + rac{70}{4} + rac{16}{4} = rac{135}{4}

Thus, the solution to the system of equations is:

(x, y) = igg(-rac{7}{2}, rac{135}{4}igg)