5x + 3y = -15

slope intercept for is: y = mx + b

3y = -5x - 15

divide both sides by 3:

**y = (-5/3)x - 5**

Lisa L.

asked • 10/12/13 What i know about my question is that is has to do with math

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5x + 3y = -15

slope intercept for is: y = mx + b

3y = -5x - 15

divide both sides by 3:

Isaak B. answered • 10/12/13

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Why? What do we usually mean when we say "solve the equation?"

There is only one equation and it contains two variables, "x" and "y".

When we say "Solve" with an equation we usually mean find specific values for the unknown variable or variables that would make the equation true.

However, the equation we have is the equation of a line.

How do I know that?

There are several types of graphs that can be created using variables x and y. Circles, lines, parabolas (ie. the path a thrown baseball takes), hyperbolas (the path of a comet or meteor that is moving too fast to be captured into an orbit, ellipses (i.e. ovals), and polynomial functions to name a few.

I know your equation is a line because the degree of each variable (the exponent) is one. If you didn't know, exponents are the tiny little numbers above and to the right of whatever has the exponent. When an exponent is one, you don't even have to show it. That's because an exponent shows how many copies of something are being multiplied together. When a number's exponent is one, only one copy of that number is being multiplied together, so nothing unusual is happening --- it's just a number. In other words, unless there is an exponent that is not one, any number's exponent is one (and we don't show it).

Example : 22 = 4 because 22 means (2)(2) = 2 times 2 = the product of 2 copies of a 2

If you are sure there is no other information or instructions, they probably want us to solve for y. The reason we might do so is to convert the equation into a different version of itself, a version that is more useful for the purpose of graphing and understanding which specific line is being described by the equation.

That version of the equation of a line is called the "slope-intercept" form of the equation of a line.

It will have the form (or in other words, "shape", or "general overall layout/ structure" or "grammar"):

y = mx + b

By that I mean x must appear only once on the right hand side, and be multiplied by a single value. That value represents the slope of the line, or the ratio of how many units of rise the line has for every unit of run. In other words, how many times more do you go up when you go a certain of number of units to the right. Example: If a line has a slope of two, that means for each unit you travel to the right along that line, you are going to rise (or go up) by two units.

b represents a number that will be added to the product of the slope and the variable x. X is a place holder for any value. If you put your finger anywhere on an x-y grid and read off the x-coordinate, and then substitute that value in wherever you see "x" in the equation and do the math, the equation in slop-intercept form will then look like m times your number plus this other number b. Doing those two operations (multiplying your number by the slope and then adding b to it) as the right hand side of the equation y = m x + b describes, will give you a number on the right hand side which is the value of y that the line passes through, in other words, the "height" or distance from the x-axis at which there is a point on the line for the value of x you specified.

If you chose x = 0, you pointed at the y-axis. In this case, mx will be the slope times zero, which is just zero, so when you add the b, all you get is 0 + b =b. That's why b is called the "y-intercept". It's the point at which the line passes through, or intercepts, the y-axis. Its the height, or distance from the x-axis, at which there is a point in the line above or below the origin. If b is positive, the line passes above the origin of the graph. If negative, the line intercepts the y-axis below the origin. If b = zero, the line passes through the origin. This is the case of equations like y = x, y = 2x, y = 3x, or y = mx (where m can be any number).

By the way, the origin is the name of the point where the x- and y axes cross over one another, drawn at the center of many graphs, unless there is some reason to focus on only some of the values. For instance, if only positive x and positive y are being considered, the origin would typically be the lower-left corner of the graph instead.

OK, how do we manipulate the equation into slope-intercept form y = mx + b?

You can do anything you want to an equation that is true, and it will still be true, as long as you've done the same thing to both sides of the equation. It is a lot like a balance weight scale. If the scales balance, they will still balance if you add or subtract the same amount of mass from both sides. For example if you weigh 100 lb and your cat weighs 20 pounds, the scale will balance if you set it to read 120 lb. If you let your cat down, you've subtracted 20 lb from your combined weight so to maintain balance if you have to also subtract 20lb from what the scale is set for, to maintain balance. It goes similarly with equations.

What do we want? We want to get y by itself on the left hand side of the equation. What is the difference between what we have and what we want? Well, y is being multiplied by 3, and something is being added. It doesn't matter what the something is, but in this case it is 5x.

Let's get rid of the something being added by subtracting it from both sides (thereby maintaining the equation's balance/weight/truth).

5x + 3y - 5x = - 15 - 5x (Some teacher's make you show this step, some do not. I'm including it for clarity)

The result after simplifying that line? (It would be ok by me to go directly here if you follow the reasoning)

3y = -5x - 15

Standard polynomial form is to list the highest degree term first, so I also switched the order of the terms on the right hand side, but this does not change what they represent any more than it would affect your combined weight whether you hold your cat in your arms or your cat climbs up onto the top of your head).

Also, writing the term with x in it first gets us one step closer to our goal : y = m x + b

Now what is the difference between what we want (y) and what we have (3y) on the left-hand-side?

Right! y is still being multiplied by 3.

To Undo that, divide both sides by 3.

3y/3 = (-5x - 15)/3 (optional step in my book, required by some teachers)

y = (-5x - 15)/3

We see the whole right side (a difference between two terms, -5x and -15) is being divided by 3 so we can distribute that division into each term in the difference).

To understand this, think of the scenario that if you had two pizzas for your three friends and your brother unexpectedly shows up and eats one of them before he realizes they aren't for him, you can think of the amount you'll each get as one-half of the half a pizza you'd originally planned for each of your friends and you to get (so one-quarter each) OR you could think of it he ate half of the original two pizzas so now you have one pizza to split among four people (so one-quarter each).

In other words (a+b)/c = a/c + b/c = (1/c)*(a + b)

y = (-5x)/3 -15/3

Or, simplifying and re-grouping a little:

y = (-5/3)*x - 5

in other words:

y = (-5/3)x + (-5)

oomph! There it is!

This equation is in the form y = mx + b.

So m is -5/3, and b is -5.

Remember what m and b meant?

m is the slope, the ratio of the rise over the run, the factor more by which you go up per amount gone to the right. So in this case, for every three units we go to the right, we are going to "rise" by a negativei five, in other words, we're going to drop five units.

Also, b = -5 means this sharply falling line (as you travel to the right) is going to pass through the y-axis at a point 5 units below the origin.

That's all there is to it. Please let me know if you have any questions!

Solve means to find all of the (x,y) pairs satisfying the equation. In this case we have

5x+3y=-15 or that 3y=-5x-15 or that y=-(5/3)x-5. For any real x, the (x,y) pair is given by

(x,-(5/3)x-5). E.g. (0,-5), (-3,0)

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