1. What is the point of intersection of 2x + 3y = 7 and 5y = 3x - 1 ?
Here are 32 lessons on simultaeous equations (aka system of equations) from Kahn Academy
https://www.khanacademy.org/math/algebra/systems-of-linear-equations
Unfortunately, you probably need to start at the beginning of Kahn Academy's algebra course to really understand this.
First of all, what is our answer going to be? A point, like (0, 0).
5th grade way: We need an x and y which make both equations
true. 2x is always going to be even. 3y will be odd only if y is odd.
We want the sum to be 7, an odd number, so y must be odd.
Let's try a 1. 2x + 3
x 1 = 7. A little arithmetic tells us x=2 since
2
x 2 + 3
x 1 = 7. So our point is (2, 1) but does this solve the
other equation: 5y = 3x -1 ?
5
x 1 = 3
x 2 - 1
5 = 6 - 1 Yes! Answer: (2, 1) If it hadn't
worked on our second equation, we would have tried y = 3 or y = -1.
If these hadn't worked we'd have to use the 7th grade way.
7th grade way: We have two equations with two variables. Let's
get rid of the y's as our first step. So let's get the same number of y's
in both equations. To do this let's multiply both sides of the first
equation by 5 and the second by 3:
2x + 3y = 7 5y = 3x - 1
10x + 15y = 35 15y = 9x - 3
15y = -10x + 35
Since both expressions on the right = 15y, they must = each other!
So 9x - 3 = -10x + 35
(add 10x to each side) 19x - 3 = 35
(Now add 3 to each side) 19x = 38
(now divide both by 19) x = 2
We have x. To get y, use either of our starting equations and
put in a 2 for x.
2x + 3y = 7
2
x 2 + 3y = 7
4 + 3y = 7
(take 4 from each side) 3y = 3
(divide each side by 3) y = 1.
So x= 2 and y = 1. Our point is (2, 1)
Third way: Graph both equations on a piece of graph paper and see
where the lines cross. To graph a line, simply pick a value for x or y
and figure out the missing one.
Example: 2x + 3y = 7.
When y = 3, 2x + 9 = 7 or 2x = -2. Hence x = -1 Plot (-1, 3)
When y = -1, 2x + -3 = 7 or 2x = 10. Hence x = 5 Plot (5, -1)
Use a ruler on your graph paper to draw a straight line thru both pts.
Now do the same for the other equation. Look where they cross -
it should be at (2, 1).
