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):
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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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