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Master the Substitution Method for Solving Linear Equations

Grade 10 Math: Master Essential Skills & Concepts

1.1 Investigating the Basic Properties of Quadratic Relations

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Characteristics of quadratic functions

Transformations of quadratic functions

Quadratic function in general form: y = ax^2 + bx + c

Converting from general to vertex form by completing the square

Shortcut: Vertex formula

Converting general form into vertex form by applying the vertex formula

Vertex Formula Application

Graphing quadratic functions: General form VS. Vertex form

Finding the quadratic functions for given parabolas

Applications of quadratic functions

2.1 Relating the Graph of y = x^2 and Its Transformations

Combining transformations of functions

Transformations of functions: Horizontal translations

Transformations of functions: Vertical translations

Reflection across the y-axis: y = f(-x)

Reflection across the x-axis: y = -f(x)

Transformations of functions: Horizontal stretches

Transformations of functions: Vertical stretches

Quadratic function in vertex form: y = a(x-p)^2 + q

3.1 Solving Quadratic Equations

Factor by taking out the greatest common factor

Factor by taking out GCF

Factor by grouping

Factoring difference of squares: \(x^2 - y^2\)

Factoring difference of squares: <katex>x^2 - y^2</katex>

Factoring difference of squares: x^2 - y^2

Factoring trinomials

Factoring Trinomials

Solving quadratic equations by factoring

Solving quadratic equations by completing the square

Using quadratic formula to solve quadratic equations

Solving a quadratic equation with TWO REAL SOLUTIONS

Solving a quadratic equation with TWO COMPLEX SOLUTIONS

Nature of roots of quadratic equations: The discriminant

Applications of quadratic equations

Completing the square

Completing the Square

1.1 Using Linear Systems to Solve Problems

Determining number of solutions to linear equations

Solving systems of linear equations by graphing

Solving systems of linear equations by elimination

Solving systems of linear equations by substitution

System of linear equations

System of linear-quadratic equations

System of quadratic-quadratic equations

2.1 Solving Problems Involving Properties of Line Segments

Distance formula: \(d = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\)

Distance formula: <katex>d = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}</katex>

Distance formula

Midpoint formula: \(M = ( \frac{x_1+x_2}2  ,\frac{y_1+y_2}2)\)

Midpoint formula

Midpoint formula: <katex>M = ( \frac{x_1+x_2}2  ,\frac{y_1+y_2}2)</katex>

Midpoint formula: <katex>M = ( \frac{x_1+x_2}2 ,\frac{y_1+y_2}2)</katex>

Slope equation: \(m = \frac{y_2-y_1}{x_2- x_1}\)

Slope equation: <katex>m = \frac{y_2-y_1}{x_2- x_1}</katex>

Slope intercept form:  y = mx + b

Slope intercept form: y = mx + b

General form:  Ax  + By  + C = 0

General form: Ax + By + C = 0

Point-slope form: y - y_1 = m(x - x_1)

Graphing linear functions using table of values

Graphing linear functions using x- and y-intercepts

Graphing from slope-intercept form y=mx+b

Graphing linear functions using a single point and slope

Word problems of graphing linear functions

3.1 Using Analytic Geometry to Verify Geometric Properties

Parallel line equation

Perpendicular line equation

Combination of both parallel and perpendicular line equations

Parallel and perpendicular lines in linear functions

1.1 Investigating Similarity and Solving Problems Involving Similar Triangles

Similar triangles

Similar polygons

Isosceles and equilateral triangles

2.1 Solving Problems Involving the Trigonometry of Right Triangles

Use sine ratio to calculate angles and sides (Sin = o / h)

Use cosine ratio to calculate angles and sides (Cos = a / h)

Combination of SohCahToa questions

Word problems relating ladder in trigonometry

Word problems relating guy wire in trigonometry

Solving expressions using 45-45-90 special right triangles

Solving expressions using 30-60-90 special right triangles

Use tangent ratio to calculate angles and sides (Tan = o / a)

3.1 Solving Problems Involving the Trigonometry of Acute Triangles

Law of sines

Ambiguous case: SSA triangles

Law of cosines

Applications of the sine law and cosine law

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Principles of Mathematics, Grade 10, Academic (MPM2D)

1. Quadratic Relations of the Form y = ax^2 + bx + c

2. Analytic Geometry

3. Trigonometry

math

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Intro

Ex. 1

<p>- Hi welcome back to this question here. So we're going to solve these two equations and try to find the x and y, but you must use these substitutions. Substitutions just mean that you either take this y over here, you plug in this y over here. Or you take this x, plug in this x over here. Or vice versa, okay. It could be upside-down. But y goes in the y, x goes in the x okay. So what I would do is, I would suggest that you want to take a look at your two equations first and you might want to pick the one that seems easier. Like the first one, if you solve for y for the first one wouldn't it be just 6x minus seven equals y, okay?<br />
Because remember you want to solve for one variable it could be the x it could be the y, but in this case I chose the y. So what I'm going to do is take this y over here and I'm going to plug it into the second equation where the y is, y goes into y. Because these two equations are equal somewhere on the graph. So their x and y, which is a coordinate, will be the same. So that's why I'm allowed to actually take this y here and plug it into another equation's y. So let's go ahead and actually solve this equation here. So after I plug it in what that means is this y will represent this whole thing here, which is 6x minus seven. So I'm going to put a 6x minus seven in the bracket, it has to be in the bracket. If you don't you're in trouble. Copy down everything and expand this, so two, six gives you 12. That's my 12 right there, and that's a 14 here and and equals to seven okay. Now look at the calculator by yourself, that's actually 12 minus nine I can do that in my head. 14 comes over here. I think that's just 21, right? So and then you get yourself x equals to just a 7. So you found your x. So after you find one of your variables you plug it in back to your original equation up here. Either the first one or the second one, doesn't matter. Let's say you do the first one, it will be six times seven minus y equals to seven. What's six times seven, six times seven is 42, so you get yourself 42 minus y equals to seven. Yes I'm using a calculator for all of these because I'm nervous when I'm in front of a camera. Thirty-five. I bring the seven over and I bring the y over to the other side. So I get y equals to 35, okay. Now if this is a coordinate on your graph, you will have to write down seven and 35, is the point that these two equations are going to intersect each other, okay. Thanks for watching.</p>


Another way to solve systems of linear equations is by substitution. We can substitute the y/x in one of the equations with the y/x in the other equation.