This week in math 10 we focused on systems of linear equations. We were taught 3 different ways of finding the possible solutions. We learned about graphing, substation ann or elimination. Using any of these 3 options will give you the same answers. The one I found easiest was elimination.
When using elimination the biggest thing you want to do is create a zero pair with either the X or Y. Elimination can use either addition or subtraction, I will be using addition in my example because I find it easier and you’re less likely to mix up your positives and negatives.
Step 1: Rearrange equations if necessary. The zero pairs may not always be visible, if that’s the case you may need to multiply either one equation or both, so that either the X or Y makes a zero pair.
Step 2: Once you have your zero pair add both equations together. (I find it easier when you place one equation directly above the other.)
Step 3: Depending on what was you zero pair was your next step will be to isolate the variable that is left.
Step 4: Once you’ve figured out one of your missing variables place that variable in to one of your equations. (I suggestion picking the easier equation.)
Step 5: Once again you will need to isolate the variable. After doing that you should have you missing X and Y variables.
Step 6: It’s always good to verify. Once you’ve found both variables place those variable in your equation, if for both equations are true equations you have done it right. (The variable must work for both equations for it to be correct.)
Example 1: (Zero pair is already given)
-2x + 3y = 4
2x + 5y = 12
Example 2: (Zero pair must be found)
5x – 3y = -1
3x + 2y = 7
Slope point to y intercept form
This week in math 10 we learned about 3 different forms of equations. Today I will be changing point slope form into y-intercept form.
(1) General form: ex. 2x + 3y + 8 = 0
(2) y – intercept form: ex. y = 2x + 5
(3) Point slope form: ex. (9,3), 4 –> 4(x-9) = y – 3
Step 1 – Distribute the number that is in front of the brackets into all of the numbers inside the brackets.
Step 2 – You want to get y alone, and in order to isolate the y variable you need to flip the number beside the y to the other side of the equal sign. (Remember whenever bring any number to the other side you must change the sign.)
Step 3 – If there are 2 constants add or subtract them together.
Relation or Function
This week in math 10 we learned how to tell whether or not we are dealing with a function or a relation just by looking at a graph, mapping diagram or a table chart.
Graph: If there are closed dots that are found vertically in the same line or there are 2 horizontal lines that that are situated one directly on top of the other that both are not considered functions, because they share the same x value and a different y value.
Function: Because both the x values and y values are different. And there are no closed dots that are found in the same vertical line.
Relation: Because there are lines that are horizontal that pass one on top or the other. Which means that they share a vertical line.
Mapping: An easy way to tell if a mapping diagram is a function is if the lines are connected to one other number and if all the x values are different. You always want to look at you x values first and if all you x values are different it’s okay for them to share the same y value. If there are 2 or more of the same x value the only way for it to remain a function is if they all have the same y value.
Function: Because, all the x values are different and they are all connected to a different why value.
Right diagram: Function, because even though 3 and 4 share 8 as their y values it’s okay because they are different x values.
Left diagram: Relation, because 5 (the input number) has 2 different y values 2 and 9.
Table chart: An easy way to tell if a table chart is a function is if the x values and y values are different. It’s the same thing as the mapping diagram, you always want to look at you x values first and if all you x values are different it’s okay for them to share the same y value. If there are 2 or more of the same x value the only way for it to remain a function is if they all have the same y value.
Function: Because all the x values and y values are different.
Relation: Because if you look at the x values there are two 1s and both those 1s have different y values. (1,2) (1,5)
This week in math 10 I learned about function notation.
- Using inputs and outputs with its name function notation is another way of giving an equation.
- When the name (which is usually represented by a letter is upfront) is directly followed by brackets with a number insider, the number within the brackets is your “x” (also known as your input)
- When the name is followed by brackets with a variable within which is also followed up by an equal sign, that means whatever follows the equal sign is your “y” (also known as your output)
Perfect Square Trinomials
This week in math 10 we learned about and how to factor perfect square trinomials.
After factoring a perfect square trinomial you will end up with 2 binomials that will be identical. When you’re identifying if the trinomial is a perfect square the first and last term will be a square number. And the middle term will be the sum of the square root of the last term multiplied by the square root of the middle term.
This week in math 10 we learned how to deal with factoring the “ugly” polynomials. For the most part, the “ugly” polynomials are trail and error, which means that you just have to do it until you get it right. This method is good to use, but it will get harder once a larger coefficient and more variables get added. There’s no need to worry once it becomes more complicated we were taught the box method which happens to help a lot.
The polynomial used as an example is an “ugly” trinomial, and the reason why is because it cannot be factored using the $latex/ x^2$ method or the perfect square method.
Box method – First, you’ll need to draw a square with 4 sections. You’ll then place the first term on the top of the left side sections and place the last term on the bottom of the right side corner, you should have 2 open spaces. At this point, the first term of the equation and the last part are dealt with which means the only term left is the middle term. In order to find the missing numbers/variables that should fill in the 2 sections, you’ll multiply the first and last coefficient. When you get the answer you’re going to need to find 2 numbers that add up to the middle term but at the same time multiply to the answer given. You then will have 2 integers that you’ll then place them with the variable to fill in the missing section. The final step you’ll need to do is factoring and make your 2 brackets and place the results in them.
This week in Math 10 we were shown easier ways to multiply binomials. By using models, area models and distributive property. However, these models are only supposed to be used for numbers with a degree of one. I personally find using area models very helpful. The reason I chose to do the area model instead of the models is mainly that it involves the least amount of drawing and it doesn’t have to be binomial.
This is how multiplying binomials using an area model works. First, you’ll need to draw a rectangle and split it into four. You place the first half of the expression on the outside of the rectangle and the second half of the expression of the side or the rectangle. You want to align the numbers in the expression so that only one term is aligned within the smaller rectangles from within the bigger rectangle. From then on you multiply the top terms by the side terms and you place the answers in the smaller rectangles. Once you’ve multiplied each term you then collect the like terms.
This week in math 10 we started a new unit on Polynomials. We focused a lot on the basics of polynomials, which are a lot of the materials we learnt in grade 9.
We went back to looking into what a polynomials are made of (a polynomial is made up of variables, terms and degrees.)
Terms: The amount of terms decide what kind of polynomial it is. Terms are separated by – or +. If an expression has 1 term it’s considered a monomial. 2 terms is binomial, 3 terms is trinomial.
Degrees: Each variable has an exponent whether it’s visible or not (no exponent is = to 1.) Finding the degree of a monomial you add the exponents together, however if the there is more than one term you look at all the exponents than you take the biggest exponent and that becomes your degree.
Multiplying and Dividing Polynomials
Using models you can easily solve the equations given. The model will be generated by quadrilaterals to generally form an upside down L, which you’ll later turn into a rectangle by drawing lines down and across. (The larger rectangle represents , a rectangle represents x and a smaller square represents a constant. And if there are more then one of the same shapes you add a coefficient. If the shapes are coloured it’s positive and if it’s not it’s negative.) To make the model you take the first part of the equation and that becomes the length of the equation, the second part becomes the width. The answer of the equation will be found by the rest of the shapes that didn’t form the upside down L.