This week in Math 10, we learnt how to solve word problems. Word problems can be tricky and it’s easy to ix up words. There are certain words that you have to pay close attention for since it can help structure the equations. Things to remember:

• Equals: other words for this can be; is, will, was, would be, cost, etc. That means that you need an equal sign.
• Switch words: switch words are where you might need to switch up the order of how the equation is. Examples of switch words would be: less than, more than, and from.

The word problem I’m showing to solve:

The length of a rectangle is five less than three times its width. If the perimeter is 38 inches, find the rectangles dimensions.

An important thing to remember is that you should always read it at least two times because the first time you should skim through it then re-read it the second but more carefully and highlight/circle the key words.

The length of a rectangle is five less than three times it width. If the perimeter is 38 inches, find the rectangles dimensions.\

The first sentence says that L equals 5 less than 3W. Since “less than” is a switch word, our equation would be L+3W-5

The second sentence says that the perimeter total is 38 inches which means the equation would be 2L+2W=38

L=3W-5

2L+2W=38

We can use substitution to find our variables since in the first equation we already have L by itself, so in the second equation we replace L.

2(3W-5)+2W=38 –> 6W-10+2W=38 –> 8W-10=38 –> 8W=38+10 –> 8W=48 –> 8W/8=48/8 –> W=6

Now that we have W, we can plug that into the first equation.

L=3(6)-5 –> L=18-5 –> L=13

Now since we have our L and W, we have to check it to make sure that we have the correct numbers.

L=3W-5 –> 13=3(6)-5 –> 13=18-5 –> 13=13

2L+2W=38 –> 2(13)+2(6)=38 –> 26+12=38 –> 38=38

It works! L=13, W=6

This week in Math 10, we learnt to solve systems using substitution. Solving by substitution is an algebraic way of solving a system. Substitution is just inserting one equation into another equation and isolating a variable.

For this example I’ll use the equations; x+4y=-3 and 3x-7y=29.

As we can see, we don’t know what x or y is. With some rearranging, we can at least figure out the equation that a variable could be. In the first equation, x+4y=-3, we can move the 4y to the other side to get x by itself. The rearranged equation would now be; x=-3-4y. Now that we know what x is, we can insert it into the second equation. It would now look like; 3(-3-4y)-7y=29.

The first thing we do is distributive property.

-9-12y-7y=29. We now put the like terms together.

-9-19y=29. Next thing is isolation the variable. For that, we move the -9 to the other side which would make it a +9. -19y=29+9.

-19y=38. to fully isolate the variable, we divide everything by -19. Our equation would be y=-38/19 which we would then need to reduce if possible. The final answer is y=-2.

Now that we have y, we need to find x. We can use the easiest equation that we have and replace y with -2.

We will replace the y in the x+4y=-3 equation.

x+4(-2)=-3

x-8=-3. Now we move the -8 to the other side to get x by itself.

x=-3+8 –> x=5.

Now that we have x &y, we have to verify the numbers in both equations.

x+4y=-3 –> (5)+4(-2)=-3 –> 5-8=-3 –> -3=-3

3x-7y=29 –> 3(5)-7(-2)=29 –> 15+14=29 –> 29=29

The numbers worked! Our ordered pair looks like; (5,-2)

This week in Math 10, we started a new unit called; Systems of Linear Relations. There are three kinds of solutions; No solution, Infinite solutions, and One solution. In an equation, in order to figure out the solution of the two equations. It should be in slope, y-intercept form. These are how the equations should be with each solution.

No Solution – The lines are parallel to each other, the same slope, different y-intercepts. (m1=m2) (b1≠b2)

One Solution – They lines have a point where they meet. The slopes are different, the y-intercepts can be the same or different. (m1≠m2) (b1≠b2) or (b1=b2)

Infinite Solutions –  The lines are the same. The same slope and the same y-intercept. (m1=m2) (b1=b2)

Some equation Examples:

No Solution – y=3x-1 , y=3x+4

One Solution – y=2x+5 , y=-x+4

Infinite Solutions – y=12x+2 , y=12x+2

Initials: CEM

This week in Math 10, we learnt how to turn point-slope form of the equation into general form of the equation.

General form is when x and y variables are on the same side and it equals zero.  The coefficient of x always has to be a positive and nothing can be a fraction.

Point-slope form is when x and y are not on the same side, and there can be fractions.

-2(x-3)=y+15 will be our point-slope form.

Now we have to turn it into general form by getting x and y on the same side. But, before we do that, we have to do distributive property first.

Now that we have distributed, we can see that x is negative which we cannot have. That means that we have to move x and +6 to the other side. Which then makes everything that we are moving, the opposite

We now put together all the like terms and in proper order

There is our general form.

Point-slope: -2(x-3)=y+16

General: 0=2x+y+9

This week in Math 10, we learnt how to find slopes without using graphs. To find slopes, we use a formula. The formula can look like;

Let’s use the coordinates (-9,10) and (-6, -2)

We could use the formula

The equation would look like this:

Now we do the subtraction on both sides

Now we have to reduce our fraction as much as possible. Both sides can evenly divide by 3.

Since the 1 is at the bottom. the result would just be -4.

Our slope for our coordinates is -4.

This week in Math 10 we focused on relations and functions and finding the difference between the two.

When trying to figure it out if it’s a function or a relation on a graph is easier since you can see it and can visualize it. On a t-chart, you have to do a little more looking to see whether it’s a relation or a function. Same goes with if you have just the coordinates.

In words, a function cannot have y-values that share an x-value but, x-values can share a y-value.

I will show examples with a t-chart and a graph.

As you can see in the relation, there are x values that are the same.

In the highlighted picture below, you can see that at the relation, there are two dots on the same x line. On a function, that’s not allowed therefore making this a relation

In the picture below which is a function, none of them have the same x-value.

This week in Math 10, we didn’t learn very much but we got to look at our midterm exam. I didn’t exactly get a good look at the questions but I got the most incorrect in the trigonometry portion. I did realize that even on my Trigonometry unit test I got the lowest mark compared to all of my other unit tests. I think that the reason for me not doing so well would be the vocabulary and the fact that I get confused by the wording of the equations sometimes. I did get at least one question wrong in each category except for factoring.

This week in Math 10, we learnt a lot of words. The reason why I chose to do this blog post on this topic was because the vocab that we learnt can be tricky to memorize. Especially when it comes to using them in graphs. I’ll also say my tips for memorizing each.

Discrete variables – Variables that are counted. The graphs have dots on them. – A way to remember is that discrete and dots both start with a “D”.

Continuous variables – Variables that are measured. The graphs have lines on them which like the dots are connected. – A way to remember is that both continuous and connected start with a “C”.

Intercepts – It where a relation crosses (or intercepts) the axes on a graph.

Domain – Domain is the input, x variable, and the independent variable. It comes first in the ordered pair and it’s used to find the x-intercept. – One way to remember is that it’s like walls at the points and they’re coming in. For the ordered pair, I remember which goes first by thinking “walls get built first”.

Range – Range is kind of like the opposite of the domain. It’s the output, y-variable, and the dependent variable. It comes second in the ordered pair and is used to find the y-intercept. – A way to remember this is that the range is like floors and ceiling coming in. Floors and ceilings are usually built after the walls.

Dependent variable – It’s a variable that depends on the independent variable.

Independent variable – A variable that is independent amongst others and doesn’t rely on the other variables in an expression or function.

This week in Math 10,  we started our Linear Relations unit. This week was mostly review from grade 9. A cartesian plane is a graph split into 4 quadrants.

Those are the quadrants and the order of their coordinates. The x-coordinates are on the horizontal line and the y-coordinates are on the vertical line. When we are looking at ordered pairs, the first number is always on the x-axis (horizontal line). The second number is always on the y-axis (vertical line). These are the letters that we have to find on the coordinates for;

First let’s find “A”. On the x-axis, it goes to the left 7 times which means that the first number is -7. On the y-axis, it goes down 8 times which means that’s -8.

* If it goes to the left, then the first number is negative. If it goes to the right then it’d be a positive. If the coordinate goes down then the second number is negative. If it goes up, then it’s a positive.

Let’s look at “C” now. It goes to the left 5 times which makes the first number -5. It then goes up 4 times which makes the number +4.

I will also show  “D” and “F”. Now for the letter “D”. It goes to the right 6 times which makes the first number +6. Then it goes down 3 times which makes the second number -3.

Now onto “F”. It doesn’t go to the left or right which automatically makes the first number a 0. It does go up by 7 which makes the second number +7.

Now you know how to list coordinates. I will also show all of the other letters’ coordinates.