Ch 3 Notes
3-1 Solving Two-Step Equations
Two step equations will have two different operations. When you solve a two-step equation, you work BACKWARDS in the Order of Operations.
Recall:
Simplify vs. Solve
When you SIMPLIFY, you are combining like terms on ONE side of the equation or in an expression.
When you SOLVE, you are moving values through inverse operations to isolate a variable.
Simplifying uses PEMDAS
Solving uses SADMEG
Example:
¾ x + 6 = 15
Start by combining any like terms (There are none in this case)
Then move values through inverse operations beginning with subtractions and addition
¾ x + 6 = 15
-6 -6 Subtract 6 from both sides. This undoes the operation of
adding 6 on the side with the variable
¾ x = 9
4/3 * ¾ x = 4/3 * 9 Multiply both sides by the reciprocal of ¾ (which is 4/3 ) this undoes the multiplication by ¾
X = 12
3-2 Solving Multi-Step Equations
Solving Multi-Step Equations is done simply by following this easy process:
1. Combine any like terms on each side separately
2. If you have a fraction, multiply everything in the equation by the denominator to get rid of the fraction
3. Distribute if needed (if you have parenthesis)
4. Combine like terms again
5. Follow SADMEG to undo the operations on the side of the variable
Example:
2x + ½ (4x + 8) – x = 32
X + ½ (4x + 8) = 32 Combine like terms
2x + (4x + 8) = 64 Multiplied by 2 to get rid of fraction
6x + 8 = 64 Combined like terms
6x = 56 Subtracted 8
X = 56/6 or 28/3 Divided by 6
Be sure to tell me what you did at each step if you don’t show your work!!! (Your answer should look like that above)
3-3 Equations With Variables on Both Sides
This is done in the same way as the last section, except you will want to move the variable to one side.
Example:
7(3 – x) = 4x – 3(x + 5) – 6x
7(3 – x) = -2x – 3(x + 5) Combined Liked Terms
21 – 7x = -2x -3x – 15 Distributed on both sides
21 – 7x = -5x – 15 Combined Like Terms
21 – 2x = -15 Added 5x to both sides to move x to left side
-2x = -36 Subtracted 21 from both sides to undo the 21
X = 18 Divided by (-2)
When solving an equation, it is possible when the variable is on both sides, for it to cancel out. This case is known as either having no solution, or having all solutions, which is called the identity.
If the variable cancels out and the remaining statement is false, the equation has No Solution. If the variable cancels out and the remaining equation is true, the solution to the equation is the Identity.
Example:
18x – 5 = 3(6x – 2)
18x – 5 = 18x -6 Distribute
-5 = -6 Subtract 18 x from both sides
No solution since the remaining statement is False
Example:
14 – (2x + 5) = -2x + 9
14 -2x -5 = -2x + 9 Distribute the left side
9 – 2x = -2x + 9 Combine Like terms on left
9 = 9 Add 2x to both sides
Identity Since the remaining statement is true
3-4 Ratio and Proportions
A RATIO is just a comparison of two numbers in the same unit, by division. If the denominator is one, and the situation calls for a ratio, you need to keep the one on the bottom.
A RATE is a comparison of two number with different units, by division.
A UNIT RATE is a rate whose denominator is one. These are simple to figure, just divide the numerator by the denominator and keep the units.
Example:
If you can drive 187 miles on 16 gallons of gas, what is your unit rate of miles/gal?
Take 187 ÷ 16 and you get 11.69 miles per gallon.
Dimensional Analysis – the converting of units
Change 45 miles per hour into feet per second.
Start with the unit given
You have two conversions to make so you will multiply by two fractions.
Which ever unit is on the bottom is on the top in your conversion fraction, and which ever is on the top is on the bottom in the conversion fraction. Top and bottom units cancel out just like variables do.
45 miles 5280 feet 1 hour 237600 ft 3960 ft
Hour * 1 mile * 60 Minutes = 60 minutes = min
Solving Proportions
To solve a proportion, we cross multiply
a/b = c/d
a and d are known as the extremes while b & c are known as the means. The product of the means equals the product of the extremes. ad and bc are known as the cross products of the proportion
Example:
½ = 6/x
1 * x = 2 * 6
X = 12
3-5 Proportions and Similar Figures
Similar Figures have the same shape, but not necessarily the same size.
The symbol ~ means similar.
To find the value of a missing piece of similar figures, set up a proportion using the values of two corresponding pieces that you do know the value of.
Example:
If you want to find the height of a flag pole, but all you have is the length of it’s shadow, you can set up a proportion by finding the length of the shadow of a figure you do know the height of, such as a fence.
Shadow of pole Height of pole
Shadow of fence = height of fence
Set up the proportion keeping units together (Shadows on the left, heights on the rights and the pole on top and the fence on the bottom) and you can solve for the missing height.
Dilations are scaled figures that have the same center. Their sides are multiplied by a scale factor.
3-6 Equations and Problem Solving
Many times when solving problems, you will have two unknowns. The key to solving these problems is to identify one as the unknown, and then define the other in terms of the first.
For example. If you know that Bill and Joe differ in height by 4 inches and their combined height is 128 inches, you have two unknowns. (Bill’s height and Joe’s height) Let x = Bill’s height, then Joe’s height can be written in terms of Bill’s height because we know there is a difference of 4 inches in their height. Let Joe’s height = x + 4. Then when we add Bill and Joe’s heights together we set it equal to 128 and solve.
Bill’s height + Joe’s height = 128
X + x + 4 = 128
Or 2x + 4 = 128
2x = 124
X = 62
So Bill is 62 inches and Joe then would be 66 inches. (Or the other way around since we didn’t know for sure who was taller)
Often times we work with consecutive integers. These are integers that come right after each other, or differ by one. So if the first is “x”, then the next would be “x + 1”
Consecutive even or odd integers always differ by 2, so if the first is “x”, then the next (whether it’s consecutive even or odd doesn’t matter) will be “x + 2”
Formulas you should know:
Distance = Rate * Time
Cost = Number * Price
Interest = Principle * Rate * Time
3-7 Percent of Change
When figuring the percent of change, find the difference between the old and new values and then divide by the ORIGINAL value. Change this decimal to a percentage (move the decimal over two places to the right). If the value went up, it was an increase, if it went down it was a decrease.
Example:
After Christmas I buy a tree for $50. It was originally $70. What was the percent of change?
70-50 = 20. 20 ÷ 70 = 0.2857 or 28.57% decrease since the new price was lower.
Greatest possible error = the greatest possible error is half of the smallest measurement.
If you are measuring to the nearest hundredth, (0.01) the greatest possible error is half of that, or 0.005 – simply move one more decimal place right and make it a 5, everything else is a zero.
The Percentage of Error is the greatest possible error divided by the measurement.
If I measure something to be 8. 6 feet, I measured to the nearest tenth so my greatest possible error is half that, or 0.05. Divide that value by the measurement to get:
0.05 ÷ 8.67 feet = 0.005813 move the decimal to get the percentage of 0.58%
3-8 Finding and Estimating Square Roots
A square root is a number, when multiplied by itself yields the desired number. For example, the square root of 625 is the number that multiplies by itself to give 625. (It happens to be 25) Since a negative times a negative also gives a positive, square roots can be either positive or negative. We generally only work with the positives – these are called the principle square roots.
Any number that has a whole number as its square root is called a perfect square. Examples are 9, 16, 25, 36, 49 - since they have whole number square roots (3, 4, 5, 6, 7)
To estimate a square root, you can determine which perfect squares a number is between.
Estimate the square root of 13.
13 is between 9 and 16 which are both perfect squares, thus the square root of 13 is between 3 and 4, the square roots of 9 and 16. Since 13 is closer to 16 than 9, although very close to the middle of them, we would estimate its square root to be closer to 4 than 3, so maybe at 3.6
Of course you can also do square roots on your graphing calculator.
3-9 The Pythagorean Theorem
In a RIGHT triangle, the two sides that make up the right angles are called legs. The side opposite the 90 degree angle, and always the longest side, is called the hypotenuse.
The Pythagorean Theorem states that
IF you have a right triangle, THEN Leg12 + Leg22 = Hypotenuse2
You can use this theorem to find a leg or the hypotenuse when you have the other two pieces of information.
The IF part is called the hypothesis of a conditional statement
The THEN part is called the conclusion.
The Converse of a Conditional Statement switches those two around.
The Converse of the Pythagorean Theorem states that
IF Leg12 + Leg22 = Hypotenuse2 THEN you have a right triangle.
Examples:
Find the Hypotenuse of a right triangle whose legs are 3 and 4.
32 + 42 = H2
9 + 16 = H2
25 = H2
To get rid of the square, you have to take the square root.
Sq rt of 25 = 5 and the sq rt of H2 = H
So 5 = H
Find the missing leg of a right triangle if one leg is 5 and the hypotenuse is 13
X2 + 52 = 132
X2 + 25 = 169
X2 = 144
X = 12
Some special Right Triangles:
30-60-90 Has lengths in the proportion 1: Sqrt 2 : 3
45-45-90 Has lengths in the proportion 1: 1 : sqrt 2
45-45-91
Two step equations will have two different operations. When you solve a two-step equation, you work BACKWARDS in the Order of Operations.
Recall:
Simplify vs. Solve
When you SIMPLIFY, you are combining like terms on ONE side of the equation or in an expression.
When you SOLVE, you are moving values through inverse operations to isolate a variable.
Simplifying uses PEMDAS
Solving uses SADMEG
Example:
¾ x + 6 = 15
Start by combining any like terms (There are none in this case)
Then move values through inverse operations beginning with subtractions and addition
¾ x + 6 = 15
-6 -6 Subtract 6 from both sides. This undoes the operation of
adding 6 on the side with the variable
¾ x = 9
4/3 * ¾ x = 4/3 * 9 Multiply both sides by the reciprocal of ¾ (which is 4/3 ) this undoes the multiplication by ¾
X = 12
3-2 Solving Multi-Step Equations
Solving Multi-Step Equations is done simply by following this easy process:
1. Combine any like terms on each side separately
2. If you have a fraction, multiply everything in the equation by the denominator to get rid of the fraction
3. Distribute if needed (if you have parenthesis)
4. Combine like terms again
5. Follow SADMEG to undo the operations on the side of the variable
Example:
2x + ½ (4x + 8) – x = 32
X + ½ (4x + 8) = 32 Combine like terms
2x + (4x + 8) = 64 Multiplied by 2 to get rid of fraction
6x + 8 = 64 Combined like terms
6x = 56 Subtracted 8
X = 56/6 or 28/3 Divided by 6
Be sure to tell me what you did at each step if you don’t show your work!!! (Your answer should look like that above)
3-3 Equations With Variables on Both Sides
This is done in the same way as the last section, except you will want to move the variable to one side.
Example:
7(3 – x) = 4x – 3(x + 5) – 6x
7(3 – x) = -2x – 3(x + 5) Combined Liked Terms
21 – 7x = -2x -3x – 15 Distributed on both sides
21 – 7x = -5x – 15 Combined Like Terms
21 – 2x = -15 Added 5x to both sides to move x to left side
-2x = -36 Subtracted 21 from both sides to undo the 21
X = 18 Divided by (-2)
When solving an equation, it is possible when the variable is on both sides, for it to cancel out. This case is known as either having no solution, or having all solutions, which is called the identity.
If the variable cancels out and the remaining statement is false, the equation has No Solution. If the variable cancels out and the remaining equation is true, the solution to the equation is the Identity.
Example:
18x – 5 = 3(6x – 2)
18x – 5 = 18x -6 Distribute
-5 = -6 Subtract 18 x from both sides
No solution since the remaining statement is False
Example:
14 – (2x + 5) = -2x + 9
14 -2x -5 = -2x + 9 Distribute the left side
9 – 2x = -2x + 9 Combine Like terms on left
9 = 9 Add 2x to both sides
Identity Since the remaining statement is true
3-4 Ratio and Proportions
A RATIO is just a comparison of two numbers in the same unit, by division. If the denominator is one, and the situation calls for a ratio, you need to keep the one on the bottom.
A RATE is a comparison of two number with different units, by division.
A UNIT RATE is a rate whose denominator is one. These are simple to figure, just divide the numerator by the denominator and keep the units.
Example:
If you can drive 187 miles on 16 gallons of gas, what is your unit rate of miles/gal?
Take 187 ÷ 16 and you get 11.69 miles per gallon.
Dimensional Analysis – the converting of units
Change 45 miles per hour into feet per second.
Start with the unit given
You have two conversions to make so you will multiply by two fractions.
Which ever unit is on the bottom is on the top in your conversion fraction, and which ever is on the top is on the bottom in the conversion fraction. Top and bottom units cancel out just like variables do.
45 miles 5280 feet 1 hour 237600 ft 3960 ft
Hour * 1 mile * 60 Minutes = 60 minutes = min
Solving Proportions
To solve a proportion, we cross multiply
a/b = c/d
a and d are known as the extremes while b & c are known as the means. The product of the means equals the product of the extremes. ad and bc are known as the cross products of the proportion
Example:
½ = 6/x
1 * x = 2 * 6
X = 12
3-5 Proportions and Similar Figures
Similar Figures have the same shape, but not necessarily the same size.
The symbol ~ means similar.
To find the value of a missing piece of similar figures, set up a proportion using the values of two corresponding pieces that you do know the value of.
Example:
If you want to find the height of a flag pole, but all you have is the length of it’s shadow, you can set up a proportion by finding the length of the shadow of a figure you do know the height of, such as a fence.
Shadow of pole Height of pole
Shadow of fence = height of fence
Set up the proportion keeping units together (Shadows on the left, heights on the rights and the pole on top and the fence on the bottom) and you can solve for the missing height.
Dilations are scaled figures that have the same center. Their sides are multiplied by a scale factor.
3-6 Equations and Problem Solving
Many times when solving problems, you will have two unknowns. The key to solving these problems is to identify one as the unknown, and then define the other in terms of the first.
For example. If you know that Bill and Joe differ in height by 4 inches and their combined height is 128 inches, you have two unknowns. (Bill’s height and Joe’s height) Let x = Bill’s height, then Joe’s height can be written in terms of Bill’s height because we know there is a difference of 4 inches in their height. Let Joe’s height = x + 4. Then when we add Bill and Joe’s heights together we set it equal to 128 and solve.
Bill’s height + Joe’s height = 128
X + x + 4 = 128
Or 2x + 4 = 128
2x = 124
X = 62
So Bill is 62 inches and Joe then would be 66 inches. (Or the other way around since we didn’t know for sure who was taller)
Often times we work with consecutive integers. These are integers that come right after each other, or differ by one. So if the first is “x”, then the next would be “x + 1”
Consecutive even or odd integers always differ by 2, so if the first is “x”, then the next (whether it’s consecutive even or odd doesn’t matter) will be “x + 2”
Formulas you should know:
Distance = Rate * Time
Cost = Number * Price
Interest = Principle * Rate * Time
3-7 Percent of Change
When figuring the percent of change, find the difference between the old and new values and then divide by the ORIGINAL value. Change this decimal to a percentage (move the decimal over two places to the right). If the value went up, it was an increase, if it went down it was a decrease.
Example:
After Christmas I buy a tree for $50. It was originally $70. What was the percent of change?
70-50 = 20. 20 ÷ 70 = 0.2857 or 28.57% decrease since the new price was lower.
Greatest possible error = the greatest possible error is half of the smallest measurement.
If you are measuring to the nearest hundredth, (0.01) the greatest possible error is half of that, or 0.005 – simply move one more decimal place right and make it a 5, everything else is a zero.
The Percentage of Error is the greatest possible error divided by the measurement.
If I measure something to be 8. 6 feet, I measured to the nearest tenth so my greatest possible error is half that, or 0.05. Divide that value by the measurement to get:
0.05 ÷ 8.67 feet = 0.005813 move the decimal to get the percentage of 0.58%
3-8 Finding and Estimating Square Roots
A square root is a number, when multiplied by itself yields the desired number. For example, the square root of 625 is the number that multiplies by itself to give 625. (It happens to be 25) Since a negative times a negative also gives a positive, square roots can be either positive or negative. We generally only work with the positives – these are called the principle square roots.
Any number that has a whole number as its square root is called a perfect square. Examples are 9, 16, 25, 36, 49 - since they have whole number square roots (3, 4, 5, 6, 7)
To estimate a square root, you can determine which perfect squares a number is between.
Estimate the square root of 13.
13 is between 9 and 16 which are both perfect squares, thus the square root of 13 is between 3 and 4, the square roots of 9 and 16. Since 13 is closer to 16 than 9, although very close to the middle of them, we would estimate its square root to be closer to 4 than 3, so maybe at 3.6
Of course you can also do square roots on your graphing calculator.
3-9 The Pythagorean Theorem
In a RIGHT triangle, the two sides that make up the right angles are called legs. The side opposite the 90 degree angle, and always the longest side, is called the hypotenuse.
The Pythagorean Theorem states that
IF you have a right triangle, THEN Leg12 + Leg22 = Hypotenuse2
You can use this theorem to find a leg or the hypotenuse when you have the other two pieces of information.
The IF part is called the hypothesis of a conditional statement
The THEN part is called the conclusion.
The Converse of a Conditional Statement switches those two around.
The Converse of the Pythagorean Theorem states that
IF Leg12 + Leg22 = Hypotenuse2 THEN you have a right triangle.
Examples:
Find the Hypotenuse of a right triangle whose legs are 3 and 4.
32 + 42 = H2
9 + 16 = H2
25 = H2
To get rid of the square, you have to take the square root.
Sq rt of 25 = 5 and the sq rt of H2 = H
So 5 = H
Find the missing leg of a right triangle if one leg is 5 and the hypotenuse is 13
X2 + 52 = 132
X2 + 25 = 169
X2 = 144
X = 12
Some special Right Triangles:
30-60-90 Has lengths in the proportion 1: Sqrt 2 : 3
45-45-90 Has lengths in the proportion 1: 1 : sqrt 2
45-45-91