This week we learned about how to use substitution to find the solution of two linear relations. The substitution method is essentially where you take one equasion and input it into the second one. The solution is the point where the two relations would cross if they were graphed. It is possible to have lines with one solution (where the two lines cross at one point on the graph), no solutions (if the lines are parallel and the slopes are the same) and infinite solutions (if the two lines are on top of one another and have identical equations). To find the solution using the substitution method you’d need to follow the following steps. I’ll use the equations (x+4y=-3) and (3x-7y=29) as an example.
- First you would need to isolate a variable. Select a variable from either equation that seems the easiest to work with. The variable that seems the easiest to work with in these equations is the x from the first equasion (x+4y=-3) as it does not have a base associated with it. After you have chosen your variable, you can isolate it by subtracting 4y from both sides of the equal sign. The rearranged equation should look like this: (x=-4y-3).
- Next you can take (-4y-3) and substitute it into the second equation (3x-7y=29). Since x in the first equation is equal to (-4y-3) you would replace the x in the second equation with that same answer. Now the second equation should look like this: (3(-4y-3)-7y=29).
- After that you can solve the equation. Take the 3 and use the distributive law first to get rid of the brackets. The equation then becomes (-12y-9-7y=29). Then, simplify the (-12y) and the (-7y) so that the equation becomes (-19y-9=29). After you can add 9 to both sides of the equal sign so the equation becomes (-19y=38). Finally divide both sides by (-19) to get (y=-2). Now you have the y coordinate of the solution. All we need is to find the x.
- To do that, substitute the y coordinate (-2) back into the rearranged version of the first equation (x=-4y-3) so that it becomes (x=-4(-2)-3). Solve this equation to find x. (-4(-2)) is equal to (8) and (8-3) is equal to (5). Therefore, (x=5).
Now you know that the solution of these two equations and you can write it as an ordered pair (5, -2). You can verify this by using a graphing calculator or by inputting the two numbers back into one or both of the original equations, (x+4y=-3) and (3x-7y=29). If we use the first one, it would be (5-8=-3) so we know it is accurate. You can also see on the graph showed here that the two lines cross at (5, -2).
This week we learned how to change an equation from point slope form to general form. Point slope form is where the equation should look like this: m(x-x)=y-y. This form is useful because it gives a lot of information about the equation. You can immediately see the slope of the equation (slope = m) , the ordered pair being used (the second x and y variables would be the coordinates) and it can be rearranged easily to other forms like general form and slope y intercept form. General form is not very useful but you can easily tell if the equation is linear by seeing if the highest degree is 1 (making sure the variable has no exponents higher than 1). General form must only include integers (no fractions) it’s leading coefficient must be positive and the equation must equal to 0. To change point slope form to general form you can follow the steps and use the equation 9(x-2)=y-3.
1. Add 3 on both sides of the equal sign. Our goal is to get the equation equal to 0 so we start getting rid of the y and the -3. The equation should now look like this: 9(x-2)+3=y.
2. Simplify the equation. Use distributive law to simplify the 9(x-2) so that it becomes 9x-18+3=y. Then, simplify it further by adding the -18 and 3. It should then look like this: 9x-15=y.
3. Subtract y from both sides of the equal sign. We do this to make the equation equal 0 by cancelling out the y on the right side. The y should go right after the x value. Your final answer should be: 9x-y-15=0.
As long as you remember your algebra, distributive property and BEDMAS you should be able to use these steps to rearrange a variety of equations into general form.
This week we learned how to calculate the slope of a line. Slope is a number that states how steep a line is. To calculate slope from a graph, you need to find the rise and the run. Those are numbers that describe how to get from one nice point (points on the given line that are not decimals or fractions). The formula for calculating slope is rise over run. It is the same concept if you want to calculate the slope between ordered pairs.
If you take (3,12) and (5,22) as an example, this is how you would calculate the slope.
Calculate the rise and run to get from one nice point to the next. Rise is the rate the line goes up by on the y axis and run is the rate that the line goes horizontally on the x axis. The two ordered pairs (3,12) and (5,22) are both nice points which makes it easier to calculate. Use a slightly modified formula: y1 – y2 over x1 – x2 (the 1 and 2 are refering to the ordered pairs, for example (3,12) is pair one and (5,22) is pair 2). Written out, it should be 12 – 22 over 3 – 5. This should end up becoming -10 over -2. Once you have that you divide -10 by -2 to get an exact answer for the slope. If you graph the two points and the line that passes through them you can verify if the slope is correct. It should look like this:
The slope is 5 over 1 which means to get from one “nice point” to the next closest one you would need to go up 5 units and over 1.
Something I found interesting this week is that equations can be represented in so many different ways. For example you have ordered pairs, table of values, arrow diagram, mapping notation, function notation and much more. To show what some of those listed would look like, we can use 3x+4 as an example.
Ordered pairs: (1, 7) (2, 10) (3, 13) etc.
The x in the example equation represents the input value, the first number in each ordered pair (which can also be any number you choose). What you get after solving the of the equation (in this case multiplying the input by 3 and adding 4) is the y value or output. You can then write the input and output numbers in brackets which give you ordered pairs. These are good if you want to graph the numbers on a cartesian plane.
Table of values:
X | Y
The table of values has the same numbers as the ordered pairs, only instead of matching the corresponding x and y values together in pairs, you would write all the x values together and all the y values together. This is good if you want to find the rule (3x+4) and if you want to find x intercepts (the x value when y is equal to zero) and y intercepts (the y value when x is equal to zero)
Function notation: f(x)=3x+4
For function notation, you can take the general rule (3x+4) and state the name (f) and input value (x) at the beginning. To be more specific with numbers, you can write: f(1)=7 where 1 is the input and 7 is the output. Function notation is good when you want to show that the relation is a function (only one input value per output).
There are many more ways of representing equations, each meaning the same thing but being written differently. They are critical to solving different types of problems and in different situations.
This week we learned about function notation. It is a specific way of writing functions, which is a special type of relation where every x value only has one y value.
Written out, function notation looks something like this:
In the equation F is the name of the function and x is the input value. You could input a number into the equation and you would get a specific answer after you solve it.
To write a function from a table of values where, for example the x values are 1, 2, 3 and the y values are 5, 7, 9 you would follow these steps.
First, find out the rule. To do that, select a pair of x and y values to work with (let’s say 1 and 5). Then take the difference between the each y value (for example in the equation the y values increase by 2) and multiply that by the x value you chose (2×1). Then take the new number and see what you would do to it to get the original y value chosen (2+?=5). After that, write it out as an equation: (y=2x+3)
Now that you have the rule, you can write it in function notation. Write the name of the function, (usually f) then write x in parentheses to represent an input value. Finally write out the rule (2x+3). The final form should look like this: f(x)=2x+3.
This week we reviewed how to graph inequalities on a number line. For example, if you had the equation x > 2 this is how you’d display it properly.
1. Take the number(s) in the equation and find them on the number line. In this case the number is 2.
2. Look at the symbol(s). In the equation, it states that x is larger than 2, using the > sign.
3. Draw either a filled-in circle or a hollow circle on the number. The filled-in circle means that x is greater than or equal to/ less than or equal to the number. The hollow circle means that x is either greater or less than the number. In the example equation, x is greater than 2 so you would put a hollow circle around the 2.
4. Draw an arrow stating if x is greater than or less than the number. In the equation we know x is greater than 2 so you would need to draw an arrow pointing to the right of, and along all the numbers greater than 2.
The end result should look like the picture above.
This week I learned the difference between discreet data and continuous data. Discreet data usually applies to certain numbers of objects or things that are being measured. They are usually things you can count and can not be split into pieces. Some examples of discreet data would be the number of cars in a race, number of marbles in a jar or number of people at a concert. It would not make sense to have half a person, so you can tell it is an example of discreet data. When graphing discreet data using a cartesian graph you would not connect the dots to show that the information is discreet.
Continuous data however usually applies to things that can be measured. They can be measured at different intervals and in different ways. Some examples of continuous data would be the amount of time it takes to complete a task, the distance an airplane travels or the height of a building. It is possible to have one and a half meters, which is how you can tell this is an example of continuous data. When graphing continuous data you would connect the dots to show that the information is continuous.
This week we learned how to use the box method to factor trinomials that didn’t have a leading coefficient of 1 (ugly trinomials). For me, the box method is nice because it doesn’t require me to do as much thinking in my head as the inspection method.
As an example, if you have the expression these are the steps to factor it.
1. Draw a box and split it into quarters so that there are four spaces inside.
2. Write the first term () in the top left-hand corner and write the last term (10) in the bottom right-hand corner.
3. Multiply the two numbers without the variable (30) and find the factors of that number that add up to the middle term (11).
4. Look which numbers already inside the box have corresponding factors or things in common and place them in the two leftover spots. After, add in the variable to the numbers.
5. Find out which numbers multiply to get the numbers in the box and put them on the outside spaces accordingly. It works like a multiplication table.
6. Once you’re done, write the answer in parentheses (3x+5)(x+2)
This week we learned how to factor polynomials. This means that you would take an answer to a FOIL question and find out what the question was. For example, if you take you would first write down all the factors of 49. After that, find the pair of factors that adds up to 14 (7×7). Finally you would write the answer as two binomials (x+7)(x+7).
If you have an expression with two terms () that means that if it were factored, the expression would be conjugates. For example, (x+5)(x-5) are conjugated because the two constants have opposite signs (+ and -). This also means that if you multiply out the expression the +5 and -5 would cancel out and become deep pairs.