MrE's Algebra Blog

Algebra: Chapter 8, Lesson 5, page 380.

Motion Problems

D-R-T Problems (otherwise known as DIRT problems). `D` is the distance, `R` is the rate (or speed like miles per hour [mph] or kilometers per hour [kph] or feet per second [ft/sec]) and `T` is the time. Be careful and make sure that if the rate is in mph, then the time has to be in hours. If the rate is in feet/sec, then the time has to match and be in seconds as well.

For those type of problems that have 2 people starting off, remember that the first person’s time is `t`, but the second person (who leaves later) actually has `t – x` time where x is the time delay of the second person leaving. For example if one person leaves at `t` and the second leaves 2 hours later, then the second person’s time is `t – 2`.

REMEMBER, diagrams are great to get you to understand what you are looking for. The tables in the book are also good techniques. The more pictures or diagrams you have, the better chance you have of understanding what steps you have to go through! GO SLOW!!

Here are some links to excellent D-R-T examples at the PurpleMath website. Its easier to link to these than show you the same thing!

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

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Algebra: Chapter 3, Lesson 7, page 142.

Formulas

Formulas that use more than 1 letter are often called literal equation. Use the formula and solve for the desired variable by treating all other variables as if they are constants or coefficients. Constants are separated by `+` and `−` symbols whereas coefficients are like `2x`, the `2` is a coefficient.

Just solve these like any other equation using the strategies from yesterday

1. Multiply both sides to clear fractions or decimals, if necessary.
2. Collect like terms on each side, if necessary.
3. Use the addition property to move the variable to one side and all other terms to the other side of the equation.
4. Collect like terms again, if necessary.
5. Use the multiplication property to solve for the variable.

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

Algebra: Chapter 8, Lesson 4, page 373.

Using systems of equations

Steps to solve systems of equations can be:

• Graph the equations
• Substitution of one variable in terms of the other
• Adding or subtracting the 2 equations

Before you start attacking the word problem, make sure you have a plan of attack such as:

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

Click here for a pretty good link from purplemath.com with examples.

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

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Algebra: Chapter 3, Lesson 7, page 142.

Formulas

Formulas that use more than 1 letter are often called literal equation. Use the formula and solve for the desired variable by treating all other variables as if they are constants or coefficients. Constants are separated by `+` and `−` symbols whereas coefficients are like `2x`, the `2` is a coefficient.

Just solve these like any other equation using the strategies from yesterday

1. Multiply both sides to clear fractions or decimals, if necessary.
2. Collect like terms on each side, if necessary.
3. Use the addition property to move the variable to one side and all other terms to the other side of the equation.
4. Collect like terms again, if necessary.
5. Use the multiplication property to solve for the variable.

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

Algebra: Chapter 8, Lesson 3, page 367.

Addition and Subtraction for 2 linear equations.

You can add 2 (or subtract) linear equations together so that one of the variables cancels out. An example would be:

`3x – y = 9` and `2x + y = 6`

If we line them up, one under the other, we have:

`3x – y = 9`

`2x + y = 6`

Adding them together, we see that the sum looks like `3x + 2x – y + y = 9 + 6`

or

`5x = 15`

and solving for `x` makes it `x = 3`. If `x = 3`, then we can plug it into EITHER original equation, I’ll use the second one and we can solve for `y`.

So…  `2x + y = 6`

becomes `2*3 + y = 6` or `6 + y = 6` or `y = 0`. The ordered pair solution is then `(3, 0)`!

We may sometimes have to scale (multiply) ONE OR BOTH of the equations to make one of the variables disappear. Here is a link that can help!

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

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Algebra 1a: Chapter 3, Lesson 6, page 139.

Clearing an Equation of Fractions or Decimals

In equations containing fractions, you can use the multiplication property to make the equation easier to solve. To clear the equation of fractions, multiply both sides of the equation by the least common denominator (LCD) of all the fractions in the equation. If you wish to clear the decimals in an equation, multiply both sides by the appropriate power of 10 OR move the decimal places to the left (or right as necessary) for ALL terms an equal amount (e.g., make sure you move them ALL 2 places to the left – obviously, this is the same as multiplying by 100)

Remember the steps to SOLVING EQUATIONS:

1. Multiply both sides to clear fractions or decimals, if necessary.
2. Collect like terms on each side, if necessary.
3. Use the addition property to move the variable to one side and all other terms to the other side of the equation.
4. Collect like terms again, if necessary
5. Add or subtract to isolate the variable and finally
6. Use the multiplication or division or reciprocal properties to solve for the variable.

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

Algebra: Chapter 8, Lesson 2, page 362.

Substitution Method

You can solve 2 equations by solving 1 equation for 1 variable and then substituting that equivalent expression in the other linear equation. By doing this, you eliminate one variable. Solve for the remaining variable and then substitute its value in the original equation to find the first variable.

Here is an example of 2 equations to solve, its easier that way:

`x – 2y = 6` and `3x + 2y = 4`

1. Solve the first equation for x, so … `x = 6 + 2y` (added `2y` to both sides)

2. substitute for `x` in the second equation which now looks like: `3(6 + 2y) + 2y = 4`.

3. Distribute the new equation in `y`, it now looks like: `18 + 6y + 2y = 4`

4. combining like terms we have: `18 + 8y = 4` or simplifying

`8y = 4 – 18` … or … `8y = -14` and `y` is finally = `-14/8` or we reduce it to `-7/4`!

5. With `y = -7/4`, we can use the first equation to write:

`x – 2(-7/4) = 6` … or …. `x + 7/2 = 6` … or …. `x = 5/2`!

6. The solution is then, `(x,y)` or `(5/2, -7/4)`, wow!!!!

Here are some more examples done by purplemath.com with substitution!

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

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Algebra 1a: Chapter 3, Lesson 6, page 139.

Clearing an Equation of Fractions or Decimals

In equations containing fractions, you can use the multiplication property to make the equation easier to solve. To clear the equation of fractions, multiply both sides of the equation by the least common denominator (LCD) of all the fractions in the equation. If you wish to clear the decimals in an equation, multiply both sides by the appropriate power of 10 OR move the decimal places to the left (or right as necessary) for ALL terms an equal amount (e.g., make sure you move them ALL 2 places to the left – obviously, this is the same as multiplying by 100)

Remember the steps to SOLVING EQUATIONS:

1. Multiply both sides to clear fractions or decimals, if necessary.
2. Collect like terms on each side, if necessary.
3. Use the addition property to move the variable to one side and all other terms to the other side of the equation.
4. Collect like terms again, if necessary
5. Add or subtract to isolate the variable and finally
6. Use the multiplication or division or reciprocal properties to solve for the variable.

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

Algebra: Chapter 8, Lesson 1, page 358.

Solving Systems of Equations by Graphing

A set of equations for which a common solution is sought is called a SYSTEM OF EQUATIONS. A solution of a system of 2 equations in 2 variables (x, y) is an ordered pair that makes both equations true.

Take 2 linear equations and graph them (with at least 2 points for each linear equation) and where they INTERSECT is a “SOLUTION” to BOTH equations.

Pretty simple to do, but it can be time consuming in that you have to have graph paper and a ruler and some time ….

Here is a link with LOTS of examples from purplemath.com. It goes on for 2 pages so make sure that you see them both!

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

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Algebra 1a: Chapter 3, Lesson 5, page 136.

More on Solving Equations

Solve equations by first getting all variables on the same side of the equation. Get rid of the paranthesis too (distributive property if required). Go slow and ONCE YOU HAVE YOUR ANSWER, PLUG IT BACK IN TO SEE IF IT WORKS, IF NOT, CHECK YOUR MATH FOR SIMPLE MATH OR SIGN ERRORS!

Click here for some more examples. Don’t forget to try out Nutshellmathplus.com too!

Remember the steps to SOLVING EQUATIONS:

1. Multiply both sides to clear fractions or decimals, if necessary.
2. Collect like terms on each side, if necessary.
3. Use the addition property to move the variable to one side and all other terms to the other side of the equation.
4. Collect like terms again, if necessary
5. Add or subtract to isolate the variable and finally
6. Use the multiplication or division or reciprocal properties to solve for the variable.

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

Algebra: Benchmark “#2” (Chapters 1, 2, 3, 4 and 7)

Benchmark #2 today!

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Algebra 1a: Chapter 3, Lesson 5, page 136.

More on Solving Equations

Solve equations by first getting all variables on the same side of the equation. Get rid of the paranthesis too (distributive property if required). Go slow and ONCE YOU HAVE YOUR ANSWER, PLUG IT BACK IN TO SEE IF IT WORKS, IF NOT, CHECK YOUR MATH FOR SIMPLE MATH OR SIGN ERRORS!

Click here for some more examples. Don’t forget to try out Nutshellmathplus.com too!

Remember the steps to SOLVING EQUATIONS:

1. Multiply both sides to clear fractions or decimals, if necessary.
2. Collect like terms on each side, if necessary.
3. Use the addition property to move the variable to one side and all other terms to the other side of the equation.
4. Collect like terms again, if necessary
5. Add or subtract to isolate the variable and finally
6. Use the multiplication or division or reciprocal properties to solve for the variable.

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

Algebra: Chapter 7 Review for Algebra Benchmark #2

Algebra Skills Practice 19/20 and the Benchmark #2 Practice and Review were started in class. The end of the Chapter Review is a good overview and practice of the entire chapter.

Remember, to solve problems using the slope-intercept formula, `y = mx + b`, you need to have (or solve first for) the slope, then using one of the ordered pairs given `(x, y)` solve for the y-intercept, b.

Given the slope `m`, and the y-intercept `b`, we can develop the equation by plugging in the values for the slope and the y-intercept.

An equation perpendicular to the given one will have to have its slope be the negative reciprocal for the product to be -1. In other words, `m_1 * m_2 = -1`

A new equation that has to be parallel to the given one, must have its slope be exactly the same, `m_1 = m_2`!

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Algebra: Chapter 3, Lesson 4, page 130.

Expressions and Equations

The phrase the quantity suggests a grouping of terms will follow. The words sum of, difference of , product of and quotient of also suggests a grouping of terms (USING PARENTHESES) to follow.

The quantity “3 less than a number” is written `n−3`. The text “4 times the quantity 3 greater than a number” is translated to `4(n+3)`.

Finally, here are some problem-solving guidelines:

Phase 1: UNDERSTAND THE PROBLEM

• What am I trying to find?
• What data am I given?
• Have I ever solved a similar problem?

Phase 2: Develop and carry out a PLAN

• What strategies might I use to solve the problem?
• How can I correctly carry out the strategies I select?

Phase 3: Find the ANSWER and CHECK

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

Algebra: Chapter 7 Review for Algebra Benchmark #2

Algebra Skills Practice 19/20 and the Benchmark #2 Practice and Review were started in class. The end of the Chapter Review is a good overview and practice of the entire chapter.

Remember, to solve problems using the slope-intercept formula, `y = mx + b`, you need to have (or solve first for) the slope, then using one of the ordered pairs given `(x, y)` solve for the y-intercept, b.

Given the slope `m`, and the y-intercept `b`, we can develop the equation by plugging in the values for the slope and the y-intercept.

An equation perpendicular to the given one will have to have its slope be the negative reciprocal for the product to be -1. In other words, `m_1 * m_2 = -1`

A new equation that has to be parallel to the given one, must have its slope be exactly the same, `m_1 = m_2`!

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Algebra 1a: Chapter 3, Lesson 3,  page 125 (day #2).

Using the Properties Together

You can solve these equations in a variety of ways. Remember the objective is to ISOLATE the variable on one side. You can add, subtract, multiply or divide the same thing on both sides of the equation.

Follow these 4 steps:

1. You should solve equations with parenthesis using the distributive property
2. If there are like terms on one side of the equation, collect those first.
3. Add and subtract the constants
4. Finally, multiply or divide by the coefficient (next to the variable) to isolate the variable

Practice makes perfect and be sure to SHOW ALL THE STEPS! Purplemath has some examples here about these multi-step equations.

Don’t forget to use the reciprocal, its sometimes the same as dividing but faster.

DO NOT TAKE ANY SHORTCUTS, SHOW ALL THE WORK IN ALL ITS GORY DETAIL, THIS WILL REALLY SAVE YOU IN THE LONG RUN BY CUTTING DOWN ON SILLY MISTAKES!!

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

Algebra: Chapter 7, Lesson 8, page 338.

Parallel and Perpendicular Lines

Parallel lines by definition have the same slope. So, for the equation of 2 lines, all we have to do is figure out what the slope is of them both. If they have the same slope, then they are parallel. Check too, however, to make sure that both lines have DIFFERENT y-intercepts. If they have the same slope and y-intercept, then they are the same line, one on top of the other.

Perpendicular lines are lines that intersect at 90° or are at right angles to each other. By definition, the slopes of 2 lines that are perpendicular, when multiplied together, have a resultant product of −1.

Remember, the slope-intercept formula to find the slope, `m`: `y = mx + b`

You MAY have to solve the equation lines for `y`, isolating it to see what the slope, `m`, is as well as the y-intercept, `b`.

Here is a link from purplemath too with more explanation and examples.

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

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Algebra 1a: Chapter 3, Lesson 3,  page 125.

Using the Properties Together

You can solve these equations in a variety of ways. Remember the objective is to ISOLATE the variable on one side. You can add, subtract, multiply or divide the same thing on both sides of the equation.

Follow these 4 steps:

1. you should solve equations with parenthesis using the distributive property
2. if there are like terms on one side of the equation, collect those first.
3. add and subtract the constants
4. finally, multiply or divide by the coefficient (next to the variable) to isolate the variable

Practice makes perfect and be sure to SHOW ALL THE STEPS! Purplemath has some examples here about these multi-step equations.

Don’t forget to use the reciprocal, its sometimes the same as dividing but faster.

DO NOT TAKE ANY SHORTCUTS, SHOW ALL THE WORK IN ALL ITS GORY DETAIL, THIS WILL REALLY SAVE YOU IN THE LONG RUN BY CUTTING DOWN ON SILLY MISTAKES!!

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

Algebra: Chapter 7, Lesson 7, page 333.

Fitting Equations to Data

“The mathematical relationship between 2 variables is of interest in many real-world situations. The relationship between 2 variables can often be expressed as a linear equation, which is calleda model of the situation. The model can be use to make estimates or predictions about the quantities represented by the variables.”

In the problems with real-world data, we sometimes have to approximate, by plotting the data on a x-y graph and then drawing (fitting) a line the best way we can through MOST of the data. We can then use 2 of the points on our line to use to develop our linear equation.

We can use either the slope-intercept equation (`y = mx + b`) or the point-slope equation [`y – y_1 = m(x – x_1)`] to develop our linear equation.

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!