Hello again all, Johnathan here today to explain the addition method, aka the elimination method for solving linear systems. This method is a neat and useful method for solving linear systems which I will do my best to summarize for you now. The first step to the addition/elimination method is to ensure that your equations are all in standard form, ax+by=c. The next step is to make it so that either the x or the y value in each equation have the same coefficient but opposite signs. If this is not already the case, you can multiply all numbers in one (or both) of the equations by a number that will make it happen. Once you’ve done this, the variables which now have the same coefficient and opposite signs can be crossed out and are no longer a part of the equation. Once these variables are crossed out, add together the remaining parts of both equations. But why does this work? Let’s look at a quick example: Say we have a system of equations which has already been put through step 2: 3x-2y= 5 -1(3x+y= 11) ⇾ -3x-y= -11 If we cross out the x variables which now have the same coefficient and opposite signs, we can then add the equation and solve. (-2y= 5) (-y= -11) (-2y-y)= (5-11) ⇾ -3y= -6 Another way this could be visualized is if we bring back the previously removed variables: 3x-2y= 5 -3x-y= -11 (3x-3x)+(-2y-y)= (5-11) ⇾ -3y= -6 As you can see, the x variables cancel each other out once again, in total equaling 0. Ensuring that these variables cancel out is what allows us to solve for the other variable. Either way, this limits the equation to only one variable, thus allowing us to solve for it. -3y(÷3)= (-6÷-3) y= 2 Now that you know the value of y, you can take one of the original equations and substitute the y variable for its value and thus find x, and there you have it: the elusive and beautiful Correct Answer. Hopefully. If you’d like to make sure your answer is correct, you can now substitute both values into the other original equation to make sure the math checks out. If this does not break any fundamental laws of mathematics, the math checks out and you're good. If this does break math or is 0 = 0, see below. IMPORTANT THINGS:
Hey y'all, so today, I (kiran) am going to be explaining how to a) solve a linear system by graphing and b) giving you a vocab toolkit to do so in this workbook. (Before we begin: This is my second time 'round as I accidentally deleted half of my post, so forgive me for its poor quality) To start, a system of linear equations is a set of 2 or more linear equations. (Never could of guessed) For example, 4x + y = 8, and x + 3y = 7 . In grade 10 we will only be dealing with 2 equations, so lucky us! We graph linear equations to get a visual representation of the problem so we can find the solution. (or absence of one) Let me explain: there are 3 types of solutions to a linear system. One solution, no solution, or infinite solutions. These essentially describe what we are seeing on the graph and in an equation. Here's more info below: One solution: When graphed, these two linear equations are perpendicular to each other. They only intersect at 1 point, and only ever 1 point, so they only have one shared x and y value. This gives them only one point in common which is the one solution. It is a consistent (has a solution) system of independent equations. (Independent because the two equations are completely different except for their one shared point. They do not rely on each other) No solutions: When graphed, these two linear equations are parallel to each other. They never intersect, and have no shared x and y value. This means that there is no solution. It is an inconsistent (does not have a solution) system of independent equations. Infinite solutions: When graphed, these two linear equations are literally the same line. There are an infinite number of solutions because both lines share every single x and y value. It is a consistent (has a solution) system of dependent equations. (Dependent because the two equations are the same: they rely on each other) Ok so now we know to identify the number of solutions based on a graph, but now we actually have to graph these 2 linear equations. Here's how: 1. If the equations aren't in slope-intercept ( y - y₁ = m(x - x₁)) or standard form (Ax + By = C) turn them into those 2 forms. 2. Graph them on the same grid 3. Find the intersect point of the two lines. (or don't if the two lines are the same - infinite solutions) The solution = the ordered pair (Eg. (4, 2)) of the intersect point 4. Put the ordered pair into the equation ( replace the x with the 1st number and the y with the 2nd of the ordered pair) 5. Label the intersect point (Ie. the solution) Here are some things to remember when graphing these linear systems: - Simplify your equation before starting (if you can) - With no solutions (a parallel line) the linear systems will have the same slope, but a different y-intercept - Different slopes will almost always have different y-intercepts - Infinite solution linear systems seem easy to identify, but they often do not look like the same linear equation until you solve each one. (Basically just don't guess, solve.) Anyways, I hope that made at least a little sense to you. If not, whoops. Sorry. Here's a video that includes all the options of solutions, but only solves for 1 (no solution):
Hey, Younes here in class today we have learned about function notation, which is f(x) . To put it simply, f(x) is equivalent to what we used to write as “y”, also it isn’t f multiplied by x but f of x. Ex: f(x)=mx +b is the same thing as y= mx + b. Something to remember is that once you put a number in the f(x) position, it implies the value of y depending on what x is. For example a notation, f(3) shows the value of “y” when x=3. Hello all, Annabelle here. Today our lesson was on linear applications and modelling, which is basically a fancy way of saying how to use graphs to figure out day to day problems. These would most commonly be shown as word problems and learning how to convert the information into graphs is an important concept and one that involves lots of factors. Something to consider when analyzing data to try and graph it is, "how do we decide which one is x and which one is y?" You can figure this out by asking yourself, what are the variables, and which one is dependent? The dependant variable =the y-axis and the independent variable=the x-axis. Some vocabulary to know is slope = rate of change which we all probably know already, but the units for that in word problems can be shown as 'per.' Though it can sometimes be more intuition rather than exact steps, here some rough guidelines to follow when turning word problems into graphs. 1) figure out what the question is asking. What forms are your equations supposed to be in, what information do you have and what information do you need. 2) figure out variables. what represents x and what represents y? 3) form equation based on information, this could mean calculating the slope. Again, check to see which form your equation should be in. * 4) graph *side note: you may have to do algebra to convert one form of linear equation to another. If you're struggling, i'd highly recommend watching videos on how to do this, they usually explain things step by step and I find it very helpful. Happy graphing! Holla holla guys, gals, and non-binary pals, today we're having a hoot and a holler, a true banger if you will, today we will be discussing equations of parallel and perpendicular lines as well as how to manipulate them and represent them on graphs. As you will remember from previous lessons, the slope of an equation is what must be manipulated to make lines parallel or perpendicular to each other. The difference this time is that instead of using lines on a graph, we'll be working with equations of lines in various different forms. Since the laws of geometry remain somewhat constant, similar to the last lesson when parallel and perpendicular lines were discussed, parallel lines share the same slope, and the slope of perpendicular lines is the negative inverse of the original slope (in other words, flip the sign and flip the fraction). To find lines parallel or perpendicular to those represented in equations, finding the slope of the equation you are given is going to be necessary. This is can be very simple depending on what form the equation is written in. y=mx+b (Slope-Line Intercept Form) This form has a pretty big advantage over the other ones in that you are given the slope in the original equation itself, making it remarkably easy to find the slopes of parallel and perpendicular lines. Ax+By=C (Standard Form) This from is one of the more complex ones. There are two ways to go about getting the slope of this equation. The first is to just memorize the formula for slope when using standard form (m= - A/B). The other way is to rearrange the equation using algebra. This can be done several different ways depending on how you prefer to do algebra. However, if you do plan on using this method, make sure to remove all fractions by multiplying by the lowest common multiple of the denominators before rearranging the equation. y − y₁ = m(x − x₁) (Point-Slope form) Similar to the first form, the slope of equations in standard form is also displayed in the original equation. I don’t recommend using algebra to turn equations into this from however as it is much more complex than slope-line intercept form. In lesson 5.1, we learned about different forms for the equation of a line. Here is Slope-Intercept form. First, let's talk about what Slope-Intercept form is. It's the equation of a line in the format y = mx + b. In this equation, m, the co-efficient of x, is the slope of the line, and b, the constant, is the y-intercept (0, y). This format is very useful to instantly graph the equation, because you already know the y-intercept and the slope. Plot the y-intercept and find another point using the slope*, and then draw a line through those two points. For example, let's try the line y = 3x + 2. The y-intercept is (0, 2) and the slope is 3, or 3/1. If there is no constant, then the y-intercept is (0, 0). Remember, y = 3x is the same as y = 3x + 0 Similarly with the co-efficient of x, if there is none, then the slope is 1. y = x + 2 is the same as y = 1x + 2 *If the point you would graph based on the y-intercept and slope is not on your grid, then just remember that you can go in the opposite direction! ex: 3/2 = -3/-2. If you are given the y-intercept (0, y) of a line and its slope, you can reverse-engineer that information into Slope-Intercept form without having to graph the line. For example, if you are given the y-intercept (0, -3) and the slope 1/2, you can put those directly into your format of y = mx + b, and turn it into y = 1/2x - 3. Finally, let's talk about the big gross elephant in the room, "Standard Form". Standard form is written as Ax + By = C. If you ever come across something in this form, I suggest you use algebra to turn it into Slope-Intercept form so that it is more clearly understandable, but otherwise, let's learn how this format works. In Standard Form, A, B, and C must all be integers, and A must be positive. To convert fractions into integers, multiply the whole equation by the LCM of all of the denominators. Now, there are still shortcuts to find the slope and y-intercept. In Standard form, slope = -A/B In Standard form, y-intercept = (0, C/B) So if you have 3x - 2y = 4, your shortcut to the slope is -3/-2 (3/2), and your y-intercept is (0, 4/2) or (0, 2) Hey people Katja here. The video and I are going to explain point-slope form. This is the equation: The 2 coordinates make the point: (x₁, y₁) (x₁, y₁) can be any point on the line Let’s say our: point = (4, -1) (x₁, y₁) Slope (m) = 3 1st step: substitute the given point into the equation Criss cross applesauce the coordinates in. y - (-1)= m (x - (+4)) 2nd step: Our signs need to switch The sign will always be opposite: + → - - - → + so: 4 → -4 -1→ 1 y - (-1)= m (x - (+4)) y + 1 = m (x - 4) 3rd step: Substitute slope value into the equation and get rid of the brackets m (slope)= 3 y + 1 = m (x - 4) y + 1 = 3x - 4 Ok we’re done! You can use algebra to write this in standard form and slope intercept form. Hopefully you understood this and thanks for reading! Hey, its Lilly and I'm going to talk about special cases of linear equations, which from what I understand involves writing the equations for vertical and horizontal lines, writing the equation of a line from two points and determining whether equations are perpendicular or horizontal. To start off with, horizontal equations. These are pretty easy, the formula in slope intercept form is x=k. The y value is not necessary because y could be any real number therefor irrelevant. Secondly vertical equations. The formula for these in slope intercept form is x=k. As they are opposite to horizontal equations, the x value doesn't need specification as it could also be any real number. When writing an equation from two points it is good to first determine slope, by putting delta y over delta x, then arranging one of the points into the equation along with the slope in the form of a slope point equation. Once this has been achieved the equation can be graphed or be put into another form of equation algebraically. The last thing we covered today was determining whether equations are are perpendicular, horizontal or neither. In order to determine this just find the slope of the two equations the if they are the same then the equations are parallel, if they are the opposite sign reciprocal then they are perpendicular and if they are neither then they are neither. |
Find your green dot!AuthorsWe are members of Esquimalt High School's Gifted Math 10 class. Most of us like Hi-Chews. Archives
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