Category Archives: Math 10

Week 18-Math 10

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This week we continued going through different ways to solve systems. Last week I talked about using substitution, but another way to solve a system is by using elimination.

With elimination you would have to make a zero pair, so make one term cancel out in each equation by making one negative and one positive. If one equation has for example a -2x and the other a 2x it’s easier, but if not then you would have to multiply everything by a number so that the terms would be the same but one negative and one positive, for example if you have -3x-3y=9 and 5x-3=y you would have to times everything by 5 in the first equation and everything by 3 in the second so that when you get -15x and 15x they would make a zero pair. The equations would now look like -> -15y=45 and -15=3y    Then add the y’s and the numbers together -> -12y=30 -> 12y/12=30/12  -> y=5/2 …. And then you would continue finding the x with the known y and testing it to see if it works.

More examples:

       

Week #17 in math 10

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This week in math I learned how to solve a system using substitution.

You start with two equations like x+2y=4 and 3x+2y=0. Then you would need to find the equation that looks the easiest to use as a substitution (usually the one with a variable that has a coefficient of one). Then you would rearrange the equation so that only the variable is one one side of the equation. -> x = -2y+4 and 3x+2y=0

Then you would plug in the x into the x in the second equation -> 3(-2y+4)+2y=0

Then simplify it -> -6y+12+2y=0 -> -4y+12=0 ->   -4y = -12   ->  -4y/-4= -12/-4  -> y=3

Then plug the y into the other equation to find x -> x+2(3)=4 -> x+6=4  ->  x=-2

Then plug the x and y into an equation to see if it works : -2+2(3)=4  -> (It works)

Some more examples:

     

Week #16 in Math 10

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This week in math I learned how to tell how many solutions a system can have. A system is a set of equations that you use to find the solution, and there can be either one solution, 0 solutions or infinite solutions.

When there are lines on a graph that don’t intersect there would be 0 solutions. If we use y=mx+b the slope would be the same for both equations but the y intercept would be different which means that the lines are parallel, so they wouldn’t touch anywhere.

When there are lines on the graph that intersect there would be one solution. The slopes would be different therefore the two lines would have to touch.

When there are lines that are equivalent there would be infinite solutions. The two lines have the same slope and the same y intercept therefore they are on top of each other (coincidental lines) and they would be touching forever.

 

Examples:

Week #14 Math 10

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This week I learned how to change an equation from point slope form to general form. Point slope form is useful because it tells us the slope, along with the first x and y coordinates. General form is something that we use to make an equation look nicer, but it doesn’t really give us any useful information that we can use.

Example of what point slope form can look like: 4/3(x-2)=y-3  -> The -2 is the x coordinate and the -3 is the y coordinate. We know that they were originally both positive because if they were negative we would change the negative sign to a positive sign, since two negatives make a positive. The 4/3 is the slope, and we know that it is increasing because it’s positive.

An example of what general form can look like: 2x+5y-15=0  -> When we use general form the leading coefficient always has to be positive and an integer. We need to make the x and the y go onto one side of the equal sign and have the x be the leading coefficient. It also needs to equal 0.

To change an equation from point slope form to general form we would have to use some algebraic skills to make everything go onto one side.

Examples:

 

Week #13 in Math 10

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This week I learned how to find the slope of a line on a graph. The slope is a number that describes the steepness of a line. You would first need to find the rise and the run, and that would either give you a fraction or a whole number.

Start by finding two perfect dots that you can use as your first coordinates, they would be on the very corner of a square on a lined graph. Then you would find their coordinates. For example (3, 2) and (6, 5). First you would find the rise, so the difference in y. You would look at the leftmost dot since we look at it from left to right, and compare it to the rightmost dot by seeing how much it changes. Does it go up or down? In my example my two y coordinates are 2 and 5 which means my y goes up by 3. Then you would find the run, so the x. You would do the same thing but look at the x coordinates. In my example I used 3 and 6 so it goes up by 3 again. Rise goes over run as a fraction which means my slope would be 3/3.

More examples:

Week #12 in Math 10

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This week in math I learned the difference between a relation and a function. A relation is when one input has several outputs. Like one person would have many friends. Whereas a function is when an input has only one output, like a couple. A hint is that on a graph, almost all straight lines are functions, except for vertical lines. Vertical lines are relations. You can also tell if something is a relation on a graph if you can draw a straight vertical line in between a curve or line.

Examples:

Week #11 in Math 10

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This week we got our midterms back and I wanted to write about what I learned from my mistakes. I learned that when finding what is equal to a polynomial instead of there being multiple same variables, you can just look for coefficient that has the total amount of things you are finding. For example on the test the question was what is equal to (a-b)^2 ?

This is how I would find it:

(a-b) (a-b)

a^2-ab-ab+b^2   Since there are two negative ab’s I can just do -2ab to indicate that I have two of them which I didn’t realize before: a^2-2ab+b^2

More examples:

Week #10 in Math 10

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This week I learned how to find the domain and range on a graph. The domain is the set of input values (x) and the range is the set of output values (y).

To find the domain, we start by looking from left to right (smallest value to biggest) and see where the line starts. Then, if the dot is open we would do the number<x because we know that x is bigger than the number, and we don’t do a less than or equal to sign because we know that the line starts after the dot. If the dot was closed we would do ≤. Then we would find where the lines ends and show that x is smaller than that number. So all together it would look like number<x<number or number≤x≤number.

To find the range, it’s basically the same thing except we look from the lowest part of the y value to the highest part of where the line is. You would first look at the lowest number and indicate that y is bigger so number<y and then you would look for the highest part that the line goes so the highest number. And you would show the whole thing like number<y<number. Again, if the dots are closed you would need to put the less than or equal to sign or more than or equal to sign. 

If there is an arrow that means the line doesn’t stop so we would do the infinity sign.

Examples:

 

Week #9 in Math 10

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This week in math I learned how to solve for the x intercept and the y intercept in an equation.

For example with  3x+5y=20 to find the y intercept we would put a 0 in front of the 3x which would basically mean we could rewrite the equation as 5y=20 since we got rid of the 3x.

Then, we would divide 5y by 5 and we do the same thing on the other side so we divide 20 by 5, which gives us our answer that y=4. Our final answer is in coordinate form which is (o, 4) because the y intercept is the second number in the brackets.

To find the x intercept we would do the same thing, except get rid of all the numbers with y’s in the equation. Then once you put your answer in coordinate form, your answer would be the first number in the brackets. So (x, 0).

If you are trying to find an x^2 or y^2 and before you square root it it’s a positive number, you need to put two answers because if you multiply two positive numbers together they would equal a positive and if you multiply two negatives they would also equal a positive, so your coordinate number would have to be positive and negative.

 

              -> finding the y-intercept

 

            -> finding the x-intercept

Week #8 in Math 10

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This week in math I went through different ways of multiplying polynomials and I went through the area model technique because I still wasn’t really confident in using it, but now I understand it.

First you make a box with with enough squares to put all the terms in. For example if my polynomial was (6x-4+5)(2p-12+9) I would need to have 6 boxes. Then, along the outside of the boxes I would need to put all those terms in front or above a square.

Then, I would need to put the product of the two terms touching a square in each square.

Here are some examples:

        —-> Simplified answer= -18x^3+54x^2+24x-72

 

 

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