MrE's Algebra Blog

Algebra: Chapter 7, Lesson 6, page 328.

Finding an Equation of a Line

There are 2 ways that we can find an equation of a line. However, we need to have at least 2 points `(x,y)` or 1 point `(x,y)` and the slope, `m` provided. If we have those things, we can find the equation by the 2 methods below, you choose the one you like:

A. Using the slope-intercept equation, `y=mx+b`

1. (Don’t forget, for this to work, you have to be given the slope, `m`, and at least 1 `(x,y)` point.)
2. If you have the slope, `m`, plug it in for `m` above and pick the `(x,y)` that correspond to the point given.
3. In the equation then, you have the `y`, `m` and `x` known.
4. All you have to do is solve for `b`, the y-intercept.
5. Solve for `b`, then plug in the `b` and `m` into the slope-intercept equation.

WARNING: IF you are not given the slope, then you are given 2 points. Given the 2 points, find the slope `m` with the equation `m=(y_2−y_1)/(x_2−x_1)`, then proceed as in step 2 above.

With this method, you have to solve for b, the y-intercept.

B. Using the point-slope equation (which is a derivation of the slope definition), `(y−y_1)=m(x−x_1)`

1. (Don’t forget, for this to work, you need the slope and 1 point or at least 2 points from which you can find the slope.)
2. (Notice too, that there is NO LONGER a `y_2` and `x_2`, just a `y` and `x`. LEAVE IT THAT WAY!)
3. If you have the slope `m`, use it. If you have 2 points, then find the slope – like the WARNING above.
4. Choose 1 of the `(x, y)` points to use for `(x_1, y_1)` and plug in the values that you know (`x_1`, `y_1`, and `m`).
5. Solve the equation for `y` and remember that you have to distribute on the right side!

With this method you have to you the distribution method on the right. You DO NOT find the `b` or y-intercept.

Either method works, you choose what is most comfortable for YOU!

Here is a link to both methods from purplemath.com.

Two of tonight’s homework problems solved by MrE are here! Just click it!

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Algebra 1a: Chapter 3, Lesson 1 and 2, pages 114 and 119.

The Addition and Multiplication Properties of Equality (I call these 1 STEP EQUATIONS)

You can add, subtract, multiply or divide the same number to both sides of an equation and get an equivalent equation. We call these the addition and multiplication property of equality.

If `a=b`, then `a+c=b+c` and if `a=b`, then `ac=bc`

Subtraction and division are opposites of addition and multiplication, so we have no problems there. There are lots of examples at my favorite site, Purplemath, give these a look! Remember to show ALL the STEPS and that you can do these either vertically or horizontally. DON’T TAKE SHORTCUTS!

Examples:

(Addition)

`-6 = y-8`, we add 8, the opposite of -8

`-6+8=y-8+8`, we use the addition property to add 8 to both sides of the equation and finally,

`2=y`

(and Multiplication)

`y/9=14`, we will multiply both sides by `9/1` or just `9`

`9* (y/9)=9*14`, remembering that `9/9 =1`, we have

`y=126`

Whatever you do to an equation, do the S A M E thing to B O T H sides of that equation! If its `x+7`, then subtract 7 from both sides. If its `x-6`, then add 6 to both sides. If its `5x`, then divide both sides by `5` and if its `x/3`, then multiply both sides by `3`! Always do the opposite operation in these 1 step equations.

Two of tonight’s homework problems (lesson 1) solved by MrE are here! Just click it!

Two of tonight’s homework problems (lesson 2) solved by MrE are here! Just click it!

Algebra: Chapter 7, Lesson 5, page 323.

Equations and Slope

An equation `y=mx+b` is called the slope-intercept equation of a line. The slope is `m` and the y-intercept is `b`. Without having to plot points, or make a T chart, we can easily determine the slope as the coefficient in front of the `x` variable and the y-intercept as `(0, b)`, the constant in the slope-intercept equation.

If the equation is not of the slope-intercept form, solve for `y` to isolate it, just like we have done in the past. The key is to have the `y` on one side of the equation and the `x` and its coefficient and the constant `b` on the other side. Usually, you have to add/subtract terms first, then multiply/divide by the coefficient in front of the `y`.

You can easily plot an equation. Start with the `(0, b)` y-intercept and then use the slope definition of `m=(rise)/(run)` to move up/down and then left/right on the graph paper as determined by the values of the rise and run. Remember to watch the signs of the rise and run.

For example, find the slope of:

`2x + 3y = 7`

first subtract `2x` from both sides

`2x – 2x + 3y = -2x +7`

to give us

`3y = -2x + 7`

divide both sides by `3` to isolate the `y`

`y = (-2/3)x + 7/3`

and the slope is then `(-2/3)` and the y-intercept is `(0, 7/3)`

See these examples from purplemath.com too! Here are others to help you graph equations given the slope m and the y-intercept b.

Two of tonight’s homework problems solved by MrE are here! Just click it!

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Algebra 1a: Chapter 2 BENCHMARK!

Algebra: Chapter 7, Lesson 4, page 318.

Slope of a Line

The slope of a line is the tilt or slant. The slope is defined as the letter `m` and `m=(rise)/(run)`, where the `rise` is the change in the y-coordinate and the `run` is the change in the x-coordinate. This definition works great when you have 2 points on graph paper. You can just count the squares for the rise and the run. It doesn’t matter too, which point you start with.

Remember, slope is positive if the line is going from lower left to upper right. The slope is negative if its coming from upper left to lower right.

A slope of  `0` is a horizontal line and a line with NO SLOPE is a vertical line.

Another definition of slope, given 2 point and NO GRAPH PAPER is:

`m=(y_2−y_1)/(x_2−x_1)`, where `(x_1,y_1)` and `(x_2,y_2)` are 2 points on the line.

For example, for the points (2, 3) and (-4, 2), the slope is:

`m = (2 – (-3))/(-4 – 2) = 5/(-6) = -(5/6)`

Purplemath has this link as well.

Two of tonight’s homework problems solved by MrE are here! Just click it.

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Algebra 1a: Chapter 2 Review

Daily CUMULATIVE Review 2-9, (page 19) and Mixed Review 4 (page 50), make sure you got it all down!

• Distribute (push all the way through!)
• Factor
• Combine Like Terms
• Add/Subtract/Multiply/Divide with ± numbers

Algebra: Chapter 7, Lesson 3, page 313 (day #2).

Linear Equations and their Graphs

Linear equation have to have variables with a power of 1, NO mixed variable products and NO variables in an equation in the denominator. The easiest way to plot or graph an equation is to use the x-intercept and y-intercept.

• The x-intercept of a line is the x-coordinate of the point where the line intercepts the x-axis. To do this, all we have to do is set `y=0` and solve for `x`.
• The y-intercept of a line is the y-coordinate of the point where the line intercepts the y-axis. To do this, set `x=0` and solve for `y`

The standard form of a linear equation in 2 variables is `Ax + By = C`, where A, B and C are constants.

For horizontal lines, the graph of `y = b` is the x-axis or a line parallel to the x-axis with y-intercept, `b`.

For vertical lines. the graph of `x = a` is the y-axis or a line parallel to the y-axis with x-intercept, `a`.

Two of tonight’s homework problems solved by MrE are here! Just click it.

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Algebra 1a: Chapter 2 Review

Chapter 2 Review, page 110-111, problems 2-54 EVEN, make sure you got it all down!

• Distribute (push all the way through!)
• Factor
• Combine Like Terms
• Add/Subtract/Multiply/Divide with ± numbers

Algebra: Chapter 7, Lesson 3, page 313.

Linear Equations and their Graphs

Linear equation have to have variables with a power of 1, NO mixed variable products and NO variables in an equation in the denominator. The easiest way to plot or graph an equation is to use the x-intercept and y-intercept.

• The x-intercept of a line is the x-coordinate of the point where the line intercepts the x-axis. To do this, all we have to do is set `y=0` and solve for `x`.
• The y-intercept of a line is the y-coordinate of the point where the line intercepts the y-axis. To do this, set `x=0` and solve for `y`

The standard form of a linear equation in 2 variables is `Ax + By = C`, where A, B and C are constants.

For horizontal lines, the graph of `y = b` is the x-axis or a line parallel to the x-axis with y-intercept, `b`.

For vertical lines. the graph of `x = a` is the y-axis or a line parallel to the y-axis with x-intercept, `a`.

Two of tonight’s homework problems solved by MrE are here! Just click it.

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Algebra 1a: Chapter 2 Review

Skills Practice 5 and 6 sheets, pages 13 and 14, make sure you got it all down!

• Distribute (push all the way through!)
• Factor
• Combine Like Terms
• Add/Subtract/Multiply/Divide with ± numbers

Algebra: Chapter 4 Benchmark

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Algebra 1a: Chapter 2 Review

Skills Practice 5 and 6 review, make sure you got it all down!

• Distribute
• Factor
• Combine Like Terms
• Add/Subtract/Multiply/Divide with ± numbers

Algebra: Chapter 7, Lesson 2, page 309.

Graphing Equations

We can graph an equation, by building a T-chart of values for both x and y. You can choose any values for `x ` and `y` when making your T-chart. I like to use values like 0, 1, and 2. Make them easy and try to pick AT LEAST 3 points when graphing and equation. YOU MUST USE A RULER WHEN CONNECTING THE DOTS TOO!

Sometimes, it can be easier when building the T-chart to “solve for `y`” first, this just cuts down on the workload. Solving for `y` means isolating the `y` variable to one side of the equation and keeping the constants and ALL other variables on the other side.

Here are 2 good links from purplemath.com, the first about graphing in general (lesson 1) and the second about the T-charts and lesson 2!

Two of tonight’s homework problems solved by MrE are here! Just click it

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Algebra 1a: Chapter 2, Lesson 9, page 98.

Writing Equations

Plan to write and solve an equation by:

• Can I use a variable to represent an unknown number?
• Can I represent other conditions in terms of the variable?
• Can I find equivalent expressions?
• Can I write and solve an equation?

Remember too:

• UNDERSTAND the problem
• Develop and carry out a PLAN
• Find the ANSWER and CHECK

Two of tonight’s homework problems solved by MrE are here! Just click it!

Algebra 1: Chapter 4 Review

Today, we worked on the Mixed Practice 7 and 8, pages 624 and 625 in preparation for tomorrow’s CHAPTER 4 TEST! Get your notes in order!

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Algebra 1a: Chapter 2, Lesson 9, page 98.

Writing Equations

Plan to write and solve an equation by:

• Can I use a variable to represent an unknown number?
• Can I represent other conditions in terms of the variable?
• Can I find equivalent expressions?
• Can I write and solve an equation?

Remember too:

• UNDERSTAND the problem
• Develop and carry out a PLAN
• Find the ANSWER and CHECK

Two of tonight’s homework problems solved by MrE are here! Just click it!

Algebra: Chapter 4, Lesson 6, page 194.

Reasoning Strategies

Three steps to follow:

1. UNDERSTAND the problem
2. Develop and carry out a PLAN
3. Find the ANSWER and CHECK

You can also these strategies:

• Draw a diagram
• Make an organized list
• Look for a pattern
• Try, test and revise
• Use logical reasoning
• Simplify the problem
• Write an equation
• Make a table
• Work backward!

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Algebra 1a: Chapter 2, Lesson 8, page 93.

Inverse of a Sum and Simplifying

The inverse of a SUM Property: For any rational numbers, `−(a+b)=−a+(−b)`. The additive inverse of a sum is the sum of the additive inverses.

In other words, if you have a `−` in front of a paranthesis, then just change the sign of EVERYTHING inside.

For example: `−(2a−7b−6)` becomes the opposite of each term, `−2a+7b+6`.

Another example: `3y-2-(2y-4)=3y-2-2y+4`. Combining like terms, we see the answer `=2y-4`

Two of tonight’s homework problems solved by MrE are here! Just click it!

Algebra: Chapter 4, Lesson 4, page 183 and Lesson 5, page 187.

Using the Properties Together (Lesson 4)

Inequalities of 2 steps
We do the same, again, as equalities and solve by:

• Distributing when required
• Combining like terms
• Adding or subtracting terms to isolate variables and numbers (constants)
• Multiplying or dividing to finish solving for the variable, remembering to reverse the sign of the inequality IF we multiply or divide by a NEGATIVE NUMBER.

For example:

`7x + 4 ≤ 4x + 16`

subtract `4x` from both sides, that looks like

`7x – 4x + 4 ≤ 4x – 4x + 16`

now combine like terms on the left and the right sides

`3x + 4 ≤ 16`

subtract 4 from each side

`3x + 4 – 4 ≤ 16 – 4`

combine like terms again on both sides, so that

`3x ≤ 12`

and finally divide both sides by 3

`(3x)/3 ≤ 12/3`, so that finally

`x ≤ 4`

Go slow and show all the steps! Here are some more examples from purplemath.com

Two of tonight’s homework problems for Lesson 4 solved by MrE are here! Just click it!

Using Inequalities (Lesson 5)

We learned key phrases for lesson 5 (word translation problems):

• “Less than or equal to”, “is at most”, “no more than” — ≤
• “No less than”, “at least”, “more than or equal to” — ≥
• “Is less than” — <
• “Is greater than” — >

We learned to read the problem, draw a picture or understand what is being asked of us before we start solving an equation or inequality.

Remember for 2 step inequalities, we do the same, again, as equalities and solve by:

• Distributing when required
• Combining like terms
• Adding or subtracting terms to isolate variables and numbers (constants)
• Multiplying or dividing to finish solving for the variable, remembering to reverse the sign of the inequality IF we multiply or divide by a NEGATIVE NUMBER.

Here are some keyword descriptions from purplemath.com to help us with word problems (ugh …)

Two of tonight’s Lesson 5 homework problems solved by MrE are here! Just click it!

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Algebra 1a: Chapter 2, Lesson 8, page 93.

Inverse of a Sum and Simplifying

The inverse of a SUM Property: For any rational numbers, `−(a+b)=−a+(−b)`. The additive inverse of a sum is the sum of the additive inverses.

In other words, if you have a `−` in front of a paranthesis, then just change the sign of EVERYTHING inside.

For example: `−(2a−7b−6)` becomes the opposite of each term, `−2a+7b+6`.

Another example: `3y-2-(2y-4)=3y-2-2y+4`. Combining like terms, we see the answer `=2y-4`

Two of tonight’s homework problems solved by MrE are here! Just click it!