Graph the equations x + y = 12 and y = 2x

Question

. Write the solution to this system of linear equations by typing the point that they have in common. IMPORTANT: Both your x and y values should be integers. Remember that the format for a point is (x,y). For BrainHoney's sake, no spaces please.

Answer 1

Hi Caleb,

 

I like to think of systems of simultaneous equations in terms of synonyms. In your problem, one of the equations is "y=2x," which is a mathy way of saying that "y" and "2x" are the same. In this case, "2x" is just another way of saying "y." This means that every time you see "y" in another equation, you can replace it with a "2x" without changing the meaning of the equation at all.

 

The equation "x + y = 12" seems a little intimidating because it has two unknown values (x and y). It would be a lot less intimidating if it only had one variable in it, right? Happily, thanks to the other equation, we know that another way of saying "y" is "2x." So we can just rewrite this equation as "x + 2x = 12."

 

Now that you only have one variable in the equation, you can solve for x by getting it alone on the left-hand side of the equals sign:

 

x + 2x = 12

3x = 12

x = 12/3

x = 4

 

Now that we know x = 4, we can use "4" as a synonym for "x." Plugging "4" in for "x" in the equation "y=2x" gives 

 

y = 2*4

y = 8

 

Systems of simultaneous equations can feel overwhelming because all of a sudden there are multiple variables and multiple equations, none of which have obvious answers. But by treating one equation as a synonym for a variable, you can rewrite the other equation in a simpler way without changing its meaning. 

 

Sometimes when you're given two equations, neither one of them will have a variable alone on the left-hand side. That's OK! Just pick either variable (it doesn't matter which!), and use algebra to get it alone on the left, so that you have "y=..." or "x=..." Now you have an equation that you can use as a synonym for that variable, just like we did above.