MrE's Algebra Blog

Algebra 1a: 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.

Algebra 1a: 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.

Algebra 1a: 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

Algebra: Chapter 7, Lesson 1, page 304.

Coordinates

Coordinates are defined as `(x, y)` where the x-axis runs left to right and the y-axis runs up and down. The origin is where the points `(0, 0)` exists. A fancy word for the x-axis is the abscissa and the y-axis is the ordinate. There are 4 quadrants:

• I – both `x` and `y` axis have positive value (upper right)
• II– `x` axis is negative and `y` axis is positive (upper left)
• III – both `x` and `y` axis have negative values (lower left)
• IV – `x` axis has positive value, while the `y` axis has negative value (lower right)

By substituting a coordinate pair `(x, y)` into a linear equation, we can determine if the ordered pair is a solution to the linear equation. Just substitute for `x` the value of the first of the ordered pair, and substitute for `y`, the second value of the ordered pair. If the evauation is true, then the ordered pair fits on the line.

Here is  a good links from purplemath.com, the first about graphing in general for lesson 7-1.

Martin Luther King Day – NO SCHOOL!

Algebra 1a: BENCHMARK #4 TODAY!

Skills Practice 10 and 11 are great reviews for the Benchmark. Make sure that your notes are up-to-date and all your questions answered in class.

Follow these steps and you can’t go wrong … take your time …

1. If you have an absolute value, `|x|` for the variable, remove it (ignore it) and rewrite as if it didn’t exist.
2. Multiply both sides to clear fractions or decimals, if necessary. Multiply by the LCD for fractions or by 10 or 100 for decimals.
3. Distribute (whacky arrows) if you see parenthesis `( )` or square brackets `[ ]`
4. Collect like terms on each side, if necessary.
5. Use the addition or subtraction property to move the variable to one side and all other terms and constants to the other side of the equation.
6. Use the multiplication or division or reciprocal (good for fractions) properties to solve for the variable.
7. ONLY FOR INEQUALITIES – if you multiply or divide by a negative number in step 6, DON’T FORGET TO FLIP THE SIGN OF THE INEQUALITY!
8. If this was an absolute value problem `|x|`, re-insert the absolute value and check your answers, there may be 2 of them (`±`).

Don’t forget these terms too!

• Addition: plus, sum, more than, increased by, total, in all
• Subtraction: minus, difference, less than, subtract, decreased by
• Multiplication: times, product, multiplied, each, of
• Division: dividend, quotient

and these for Inequalities and Word Problems

• `<` — is less than
• `≤` — less than or equal to, no more than, is at most
• `>` — greater than, is more than
• `≥` — greater than or equal to, is at least

Algebra 1a: 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!

Algebra, Chapter 4, Lesson 5, page 187.

Using Inequalities

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.

• Consecutive numbers are written (remember when?) as `x`, `(x+1)` and `(x+2)` or  `3x+3` if we combine like terms.
• Consecutive odd/evens are written as `x`, `(x+2)` and `(x+4)` or `3x+6` if we combine like terms too.

Remember once we have the words translated for our 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 …)

Using the Properties Together

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 are here for Chapter 4-5 as well!

Algebra 1a: Chapter 4, Lesson 4, page 183.

Using the Properties Together

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 solved by MrE are here! Just click it!