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Algebra 36 - Solving Systems of Equations by Substitution

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Deirdre is working with a function that contains the following points. These are the x values, these are y values. They ask us, is this function linear or non-linear? So linear functions, the way to tell them is for any given change in x, is the change in y always going to be the same value. For example, for any one-step change in x, is the change in y always going to be 3? Is it always going to be 5? If it's always going to be the same value, you're dealing with a linear function. If for each change in x--so over here x is always changing by 1, so since x is always changing by 1, the change in y's have to always be the same. If they're not, then we're dealing with a non-linear function. We can actually show that plotting out. If the changes in x-- we're going by different values, if this went from 1 to 2 and then 2 to 4-- what you'd want to do, then, is divide the change in y by the change in x, and that should always be a constant. In fact, let me write that down. If something is linear, then the change in y over the change in x always constant. Now, in this example, the change in x's are always 1, right? We go from 1 to 2, 2 to 3, 3 to 4, 4 to 5. So in this example, the change in x is always going to be 1. So in order for this function to be linear, our change in y needs to be constant because we're just going to take that and divide it by 1. So let's see if our change in y is constant. When we go from 11 to 14, we go up by 3. When we go from 14 to 19, we go up by 5, so I already see that it is not constant. We didn't go up by 3 this time, we went up by 5. And here, we go up by 7. And here, we're going up by 9. So we're actually going up by increasing amounts, so we're definitely dealing with a non-linear function. And we can see that if we graph it out. So let me draw-- I'll do a rough graph here. So let me make that my vertical axis, my y-axis. And we go all the way up to 35. So I'll just do 10, 20, 30. Actually, I can it do a little bit more granularly than that. I could do 5, 10, 15, 20, 25, 30, and then 35. And then our values go 1 through 5. I'll do it on this axis right here. They're not obviously the exact same scale, so I'll do 1, 2, 3, 4, and 5. So let's plot these points. So the first point is 1, 11, when x is 1, y is 11. This is our x-axis. When x is 1, y is 11, that's right about there. When x is 2, y is 14, that's right about there. When x is 3, y is 19, right about there. When x is 4, y is 26, right about there. And then finally, when x is 5, y is 35, right up there. So you can immediately see that this is not tracing out a line. If this was a linear function, then all the points would be on a line that looks something like that. That's why it's called a linear function. In this case, it's not, it's non-linear. The rate of increase as x changes is going up.

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Recognizing linear functions (video) | Khan Academy

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Deirdre is working with a function that contains the following points. These are the x values, these are y values. They ask us, is this function linear or non-linear? So linear functions, the way to tell them is for any given change in x, is the change in y always going to be the same value.

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Learn what linear equations are

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If you're learning about graphs, you're bound to see a bunch of linear equations, so it's a good idea to understand what makes an equation a linear equation. This tutorial explains linear equations and shows you the difference between equations that are linear and ones that are not. Check it out!

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Virtual Nerd's patent-pending tutorial system provides in-context information, hints, and links to supporting tutorials, synchronized with videos, each 3 to 7 minutes long. In this non-linear system, users are free to take whatever path through the material best serves their needs. These unique features make Virtual Nerd a viable alternative to private tutoring.

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Introduction to function inverses | Functions and their graphs | Algebra II | Khan Academy

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Algebra Basics: What Are Functions? - Math Antics

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The substitution method | Systems of equations | 8th grade | Khan Academy

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https://mathbitsnotebook.com/Algebra1/Functions/FNNotationEvaluation.html

https://sciencing.com/ways-tell-something-function-8602995.html

https://www.youtube.com/watch?v=m8Dv8R_IeVI

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https://www.varsitytutors.com/hotmath/hotmath_help/topics/rate-of-change

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This video explains delta y and delta x. This should help you if you don't understand.

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Standard form for linear equations | Algebra I | Khan Academy

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Slope Intercept Form y=mx+b, Point Slope & Standard Form, Equation of Line, Parallel & Perpendicular

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Point Slope Form - MathHelp.com - Algebra Help

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https://www.youtube.com/watch?v=4il4haYASys

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Find the Equation of a Line Given Two Points

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Graphing Linear Equations In Slope Intercept and Standard Form - Algebra 1 & 2 Review

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https://www.sanjuan.edu/cms/lib8/CA01902727/Centricity/Domain/7044/26%20-%20Linear%20Equation%20Word%20Problems.pdf

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https://cdn.kutasoftware.com/Worksheets/PreAlg/Graphing%20Lines%20in%20Slope-Intercept%20Form.pdf

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https://cdn.kutasoftware.com/Worksheets/Alg1/Writing%20Linear%20Equations.pdf

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https://www.lcps.org/cms/lib/VA01000195/Centricity/Domain/861/Intercepts.pdf

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http://marstellerms.pwcs.edu/common/pages/DisplayFile.aspx%3FitemId%3D1513784

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http://www.wallace.ccfaculty.org/book/4.2%20Substitution.pdf

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Solving Systems of Equations... Elimination Method (NancyPi)

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Addition elimination method 1 | Systems of equations | 8th grade | Khan Academy

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https://www.gvsd.org/cms/lib02/PA01001045/Centricity/Domain/605/6-3%20solving%20by%20elimination.pdf

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https://www.youtube.com/watch?v=unSBFwK881s

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https://www.youtube.com/watch?v=hLWeHfGemBU

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https://www.youtube.com/watch?v=CA4S7S-3Lg4

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https://www.youtube.com/watch?v=FWbcKade3rw&disable_polymer=true

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https://ipadthinker.com/math/snowboardSlopeInt.html

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https://www.math-play.com/slope-intercept-game.html

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https://www.brainpop.com/games/graphinglines/?topic_id=

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Do all of the units for linear relationships

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Unit 3: Linear relationships | Khan Academy

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If you're seeing this message, it means we're having trouble loading external resources on our website.

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A linear equation is an equation for a straight line

These are all linear equations:

y = 2x + 1

5x = 6 + 3y

y/2 = 3 − x

Let us look more closely at one example:

Example: y = 2x + 1 is a linear equation:

The graph of y = 2x+1 is a straight line

When x increases, y increases twice as fast, so we need 2x

When x is 0, y is already 1. So +1 is also needed

Here are some example values:

Check for yourself that those points are part of the line above!

Different Forms

There are many ways of writing linear equations, but they usually have constants (like "2" or "c") and must have simple variables (like "x" or "y").

Examples: These are linear equations:

But the variables (like "x" or "y") in Linear Equations do NOT have:

Examples: These are NOT linear equations:

y ² − 2 = 0

3√x − y = 6

x ³/2 = 16

Slope-Intercept Form

The most common form is the slope-intercept equation of a straight line:

Example: y = 2x + 1

Point-Slope Form

Another common one is the Point-Slope Form of the equation of a straight line:

y − y ₁ = m(x − x ₁)

Example: y − 3 = (¼)(x − 2)

It is in the form y − y₁ = m(x − x₁) where:

General Form

And there is also the General Form of the equation of a straight line:

Example: 3x + 2y − 4 = 0

It is in the form Ax + By + C = 0 where:

There are other, less common forms as well.

As a Function

Sometimes a linear equation is written as a function, with f(x) instead of y:

And functions are not always written using f(x):

The Identity Function

There is a special linear function called the "Identity Function":

And here is its graph:

It makes a 45° (its slope is 1)

It is called "Identity" because what comes out is identical to what goes in:

Constant Functions

Another special type of linear function is the Constant Function ... it is a horizontal line:

f(x) = C

No matter what value of "x", f(x) is always equal to some constant value.

Using Linear Equations

You may like to read some of the things you can do with lines:

519, 2074, 1158, 2075, 520, 1159, 2455, 2456, 2457, 2458

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A linear equation is an equation for a straight line y = 2x + 1 5x = 6 + 3y y/2 = 3 − x Let us look more closely at one example: The graph of y = 2x+1 is a straight line When x increases, y increases twice as fast, so we need 2x When x is 0, y is already 1.

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Linear Equations - MathBitsNotebook (A1 - CCSS Math)

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Algebra 1 CCSS Lessons and Practice is a free site for students (and teachers) studying a first year of high school algebra under the Common Core State Standards.