3.4 The “generic” simultaneous equations

Here are the generic simultaneous equations .

• $\text{ax}+\text{by}=e$
• $\text{cx}+\text{dy}=f$

I call them “generic” because every possible pair of simultaneous equations looks exactly like that, except with numbers instead of $a$ , $b$ , $c$ , $d$ , $e$ , and $f$ . We are going to solve these equations.

Very important!!!

We can solve this by elimination or by substitution. I am going to solve for $y$ by elimination. I will use all the exact same steps we have always used in class.

Step 1: make the coefficients of x line up

To do this, I will multiply the top equation by $c$ and the bottom equation by $a$ .

$\text{acx}+\text{bcy}=\text{ec}$

$\text{acx}+\text{ady}=\text{af}$

Step 2: subtract the second equation from the first

This will make the $x$ terms go away.

$\begin{array}{}\text{acx}+\text{bcy}=\text{ec}\\ \underline{-(\text{acx}+\text{ady}=\text{af})}\\ \text{bcy}-\text{ady}=\text{ec}-\text{af}\end{array}$

Step 3: solve for y

This is something we’ve done many times in class, right? First, pull out a $y$ ; then divide by what is in parentheses.

$\text{bcy}-\text{ady}=\text{ec}-\text{af}$

$y=(\text{bc}-\text{ad})=\text{ec}-\text{af}$

$y=\frac{\text{ec}-\text{af}}{\text{bc}-\text{ad}}$

So what did we do?

We have come up with a totally generic formula for finding $y$ in any simultaneous equations. For instance, suppose we have…

$\mathrm{3x}+\mathrm{4y}=\text{18}$

$\mathrm{5x}+\mathrm{2y}=\text{16}$

We now have a new way of solving this equation: just plug into $y=\frac{\text{ec}-\text{af}}{\text{bc}-\text{ad}}$ . That will tell us that $y=\frac{(\text{18})(5)-(3)(\text{16})}{(4)(5)-(3)(2)}=\frac{\text{90}-\text{48}}{\text{20}-6}=\frac{\text{42}}{\text{14}}=3$

Didja get it?

Here’s how to find out.

• Do the whole thing again, starting with the generic simultaneous equations, except solve for $x$ instead of $y$ .
• Use your formula to find $x$ in the two equations I did at the bottom (under “So what did we do?”)
• Test your answer by plugging your $x$ and my $y=3$ into those equations to see if they work!