Solving One-Step Equations - wsfcs.k12.nc.us
Solving One-Step Equations |Solving Multi-Step Equations | |
|Multi-Step Equations w/ Distributive Property |Multi-step Equations with Variables on Each Side |
|Percent of Change |Solving Equations with Formulas |
|Relations | |
|Midpoint of a Line |Linear Equations |
|How To Graph a Line |Conversions Between Standard Form and Slope-Intercept Form |
|Finding the Slope of a Line | |
|Parallel and Perpendicular Lines |Line of Best Fit |
|Polynomial Operations |Power to a Power |
|Multiplying a Polynomial by a Monomial |binomials times binomials |
end
2/8/2012
Solving One-Step Equations
|4 + ( = 10 |Algebra |
|( ( ( ( ( | |
|( ( ( ( ( ( |Concrete example |
|( ( ( | |
|4 + x = 10 |Algebra with a variable |
|To solve: | 4 + x = 10 |(“=” means “same as”) |
| |-4 - 4 | |
| | 0 + x = 6 |This means that 6 is the only number that can replace x to make the equation true. |
| |x = 6 | |
ALWAYS SHOW YOUR WORK!!
|NO! |YES! |ASK YOURSELF: What is happening to the variable? How can I get the |
|6 = X + 2 |6 = X + 2 |variable by itself? |
|4 = X |-2 -2 | |
| | 4 = X | |
Examples: Addition/Subtraction
Notice: In these examples, you are moving an entire quantity (the constant in these examples) to the other side of the equation.
|1) | 27 + n = 46 |2) |-5 + a = 21 |
| |-27 -27 |Remember that whatever you |+5 + 5 |
| | 0 + n = 19 |do on one side of the equation you have to do on | |
| |n = 19 |the other! |a = 26 |
Examples: Multiplication/Division
Notice: In these examples, you are separating the quantity (the coefficient and the variable).
|3) |- 8n = -64 |4) | m = 12 |
| | |Remember that whatever you do on one side of the |7 |
| |-8n = -64 |equation you have to do on the other! | |
| |-8 -8 | |7 * m = 12(7) |
| | | |1 7 |
| |n = 8 | | |
| | | |m = 84 |
|5) | 5b = 145 |6) | -13 = m |
| | | |-5 |
| |5b = 145 | | |
| |5 5 | |(-13)(-5) = m * -5 |
| | | |-5 |
| |b = 29 | |65 = m |
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2/9/2012
Solving Multi-Step Equations
Compare Equations
|Equation |Comments |Equation 2 |
| X + 3 = 15 |( Total is the same ( | 2x + 3 = 15 |
| |Variable has a coefficient greater than 1 ( | |
|X + 3 = 15 | |2x + 3 = 15 |
|-3 -3 | |-3 = -3 |
| X = 12 |( One step | 2x = 12 |
| | | |
| | |2x = 12 |
| | |2 |
| | | |
| |Two steps ( |X = 6 |
Coefficient = the number in front of a variable
Constant = a number. Example: in 2x + 3, 3 is the constant.
Like Terms = quantities with the same variable (eg 2x, 0.5x, 644x, 342909835x, 1/2x, etc, are like terms because they all have an x). Numbers without a variable (constants) are also like terms!
How to Solve Multi-Step Equations:
1. Combine like terms on the left or right if you can.
2. Get all your variables on one side and all your constants on the other side (addition/subtraction OR multiplication/division)
3. Divide both sides by the coefficient of the variable.
Examples:
|A) | 7 – y – y = -1 |
|1. Combine like terms. |7 – 2y = -1 |
|2. Get all variables on one side and all constants on the other. | 7 – 2y = -1 |
| |-7 -7 |
| |-2y = -8 |
|3. Divide both sides by the coefficient of the variable. |-2y = -8 |
| |-2 -2 |
| | |
| |y = 4 |
|B) | | 13 = 5 + 3b - 13 |
|1. Combine like terms. | |13 = 3b - 8 |
|2. Get all variables on one side and all constants on the | | 13 = 3b – 8 |
|other. | |+8 +8 |
| | |21 = 3b |
|3. Divide both sides by the coefficient of the variable. | |21 = 3b |
| | |3 3 |
| | | |
| | |7 = b |
|C) | a – 18 = 2 |z + 10 = 2 |
| |5 |9 |
|1. Combine like terms. | | |
|2. Get all variables on one side and all constants on the | a – 18 = 2 |9(z + 10) = 2(9) |
|other. |5 |9 |
| |+18 +18 | |
| |a = 20 |z + 10 = 18 |
| |5 | |
| | | z + 10 = 18 |
| | |-10 -10 |
| | |z = 8 |
|3. Divide both sides by the coefficient of the variable. | a * 5 = 20 * 5 | |
| |5 | |
| | | |
| |a = 120 | |
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2/10/2012
Solving Multi-Step Equations That Include the Distributive Property
| |Example: 2 = -2(n – 4) |
|Step 1: Distribute First |2 = -2(n) + (-2)(-4) |
| |2 = -2n + 8 |
|Step 2: Solve as you would any other multi-step equation | - 8 = -2n – 8 |
| |- 6 = -2n |
| | |
| |-6 = -2n |
| |-2 -2 |
| | |
| |3 = n |
|-5(x + 3) = -45 | |
|-5x + (-15) = -45 | |
|+15 +15 |This means that 6 is the only number that you can substitute to make this equation true. |
|-5x = -30 |( |
|x = 6 | |
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2/14/2012
Solving Multi-Step Equations with Variables on Each Side
|Step 1: Add like terms (there are none) |Example: 6 – x = 5x + 30 |
|Step 2: Distribute (none to do) |- 5x -5x |
|Step 3: Variables on the left |6 – 6x = 30 |
|Step 4: Numbers (constants) on the right | -6 -6 |
| |-6x = 24 |
| |-6 -6 |
| | |
| |x = 4 |
|Step 1: Add like terms (there are none) |Example: 5x + 2 = 2x - 10 |
|Step 2: Distribute (none to do) |- 2x -2x |
|Step 3: Variables on the left |3x + 2 = - 10 |
|Step 4: Numbers (constants) on the right | -2 -2 |
| |3x = -12 |
| |3 3 |
| | |
| |x = 4 |
| |Example: 5y – 2y = 3y + 2 |
|Step 1: Add like terms |3y = 3y + 2 |
|Step 2: Distribute (none to do) |- 3y -3y |
|Step 3: Variables on the left (if possible) |0 = 2 |
|Step 4: Numbers (constants) on the right |0 does not equal 2. It is a false statement. This means that there are no |
| |real numbers that can be substituted for y in this equation. |
|Step 1: Add like terms (there are none) |Example: 2y + 4 = 2(y + 2) |
|Step 2: Distribute |2y + 4 = 2(y) + 2(2) |
| |2y + 4 = 2y + 4 ( |
|Step 3: Variables on the left |-2y -2y |
| |0 + 4 = 0 + 4 ( |
|Step 4: Numbers (constants) on the right |Both sides of the equation are equal without a variable. Therefore, any real |
| |number can be substituted for y, and the equation will be true. |
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2/16/12
Percent of Change
|Types of INCREASES |Types of DECREASES |
|New = original(1 + r) |New = original(1-r) |
|* Tax (for example, on clothing purchases) |* sales |
|* Interest (such as on a bank account) |* discounts |
|* inflation |* depreciation |
|* commission (such as when a car salesman earns a certain % of what he sells) | |
When solving a Percent of Change problem:
| | |EXAMPLE: original: 18 new: 10 |
|Step 1: |WRITE OUT THE FORMULA!!!! | New = original(1-r) | |
| |If you don’t it will be counted as wrong! | | |
|Step 2: |Organize your information. | |N =10 |
| | | |O = 18 |
| | | |r = ? |
|Step 3: |Plug your values into the equation. | 10 = 18(1-r) | |
|Step 4: |Solve the equation. | 10 = 18 – 18r | |
| | |-18 = -18 | |
| | |- 8 = - 18r | |
| | |-18 -18 | |
| | | | |
| | |.4444 = r | |
| | |44% = r | |
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2/21/2012
Solving Equations with a Formula
| | |EXAMPLE: P = 2L + 2W |
| | |If the length is 5 and the perimeter is 24, what is the value of W?|
|Step 1: |WRITE OUT THE FORMULA!!!! |P = 2L + 2W | |
|Step 2: |Organize your information. | |P = 24 |
| | | |L = 5 |
| | | |W = ? |
|Step 3: |Plug your values into the equation. | 24 = 2(5) + 2W | |
|Step 4: |Solve the equation. | 24 = 10 + 2W | |
| | |-10 =-10 + 2W | |
| | |14 = 2W | |
| | |2 2 | |
| | | | |
| | |7 = W | |
| | |EXAMPLE: P = 2L + 2W |
| | |If the width is 2.6 and the perimeter is 12.4, find L. |
|Step 1: |WRITE OUT THE FORMULA!!!! |P = 2L + 2W | |
|Step 2: |Organize your information. | |P = 12.4 |
| | | |L = ? |
| | | |W = 2.6 |
|Step 3: |Plug your values into the equation. | 12.4 = 2L + 2(2.6) | |
|Step 4: |Solve the equation. | 12.4 = 2L + 5.2 | |
| | |- 5.2 -5.2 | |
| | |7.2 = 2L | |
| | |2 2 | |
| | | | |
| | |3.6 = L | |
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RELATIONS
Domain = x values
Range = y values
f(x) = 2x2 + 7x – 2
f(2a) = 2(2a)2 + 7(2a) – 2
= 2(4a)2 + 14a – 2
= 8a2 + 14a – 2
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Midpoint of a Line
The Midpoint of a Line is the middle, center, or halfway point of a line. Both segments of the line are equal distances.
Midpoint = m x + x , y + y
2. 2
Example: Find the midpoint between (2, -4) and (-10, -3)
1. Identify your x and y coordinates: x y x y
(2, -4) (-10, -3)
2. Put into the formula: 2 + (-10) , -4 + (-3)
2 2
-8 , -7
2 2
(-4, -7/2) or (-4, -3.5) or (-4, -3½)
Example: Using the midpoint formula, we will find the missing information:
Given endpoint (-3, -7)
Midpoint (-8, 4)
Find endpoint (x, y)
Solving by formula:
x + x = xm , y + y = ym
2 2
2(-3 + x) = -8(2) , 2(-7 + y) = 4(2)
2 2
-3 + x = -16 , -7 + y = 8
+3 +3 +7 +7
x = - 13 , y = 15
(-13, 15)
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3/12/2012
Linear Equations
* What makes an equation a linear equation? Linear = makes a line
* What does linear look like?
1. No multiplication between variables together (in other words, no 2 variables are multiplied together)
2. No exponents greater than 1
| |Equation |Linear or Non-Linear |Explanation |
|A. |2x = 3y + 1 |Linear |No multiplication between variables; no exponents |
| | | |> 1 |
|B. |4xy + 2y = 7 |Non-linear |xy |
|C. |2x2 = 4y - 3 |Non-linear |x2 |
|D. |X – 4y = 2 |Linear |No multiplication between variables; no exponents |
| |5 5 | |>1 |
Linear equations can be written in standard form (Ax + By = C).
X-intercept –A point (x, y) where a line crosses the x-axis (x, 0)
Y-intercept – A point (x, y) where a line crosses the y-axis (0, y)
How do you graph a line?
1. Standard Form (Ax + By = C): use x and y intercepts
2. Slope-Intercept Form (y = mx + b): use graphing calculator:
y =
2nd graph
Plot
1. Graphing a line using Standard Form
Plot the x-intercept
Plot the y-intercept
| |Example: Graph 2x + 5y = 20 | |
| | | |
|Write your equation twice. |2x + 5y = 20 |2x + 5y = 20 |
|Cover (mark out) the x and its coefficient in one; | | |
|cover (mark out) the y and its coefficient in the |5y = 20 |2x = 20 |
|other |5 |2 |
|Solve. | | |
|Write your ordered pairs. |y = 4 |x = 10 |
|Plot your points on a graph and draw your line |x-intercept: (0, 4) |y-intercept: (10, 0) |
|through the points. | | |
| |Example: Graph 3x + y = -1 | |
| | | |
|Write your equation twice. |3x + y = -1 |3x + y = -1 |
|Cover (mark out) the x and its coefficient in one; | | |
|cover (mark out) the y and its coefficient in the |y = -1 |3x = -1 |
|other | |3 |
|Solve. |x-intercept: (0, -1) | |
|Write your ordered pairs. | |x = -1/3 |
|Plot your points on a graph and draw your line | |y-intercept: (-1/3, 0) |
|through the points. | | |
| |Example: Graph x – y = -3 | |
| | | |
|Write your equation twice. |x - y = -3 |x - y = -3 |
|Cover (mark out) the x and its coefficient in one; | | |
|cover (mark out) the y and its coefficient in the |-y = -3 |x = -3 |
|other |You can’t have a –y, so multiply both sides by -1: |y-intercept: (-3, 0) |
|Solve. |-y(-1) = (-3)(-1) | |
|Write your ordered pairs. |y = 3 | |
|Plot your points on a graph and draw your line |x-intercept: (0, 3) | |
|through the points. | | |
2. Graphing a line using Slope-Intercept Form
y = mx + b
On your calculator, press:
Y =
2nd
graph (table)
Pick out 3 or 4 points from the table that are easy to plot (ie X and Y are whole numbers) and plot them on a graph. Draw your line through those points.
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3/14/12
Conversions Between Standard Form and Slope-Intercept Form
1. Converting from Standard Form to Slope-Intercept Form
Ax + By = C ( y = mx + b
Example: Convert 4x + y = -2 to Slope-Intercept Form
|Write your original formula. | 4x + y = -2 – 4x |NOTE: You are not separating the coefficient from the variable (the x |
|Get y by itself. |-4x |from the 4), so you are not dividing both sides by 4. You are moving the|
| |0 |entire quantity of 4x to the other side of the equation, which is why you|
| |y = -4x -2 |subtract. In other words, you are moving the variable, not isolating it. |
| | |NOTE: Write the sentence so that it follows the form. Therefore, you |
| | |write y = -4x – 2, rather than y = -2 – 4x. |
Example: Convert 5x – 3y = -6 to Slope-Intercept Form
|Write your original formula. | 5x – 3y = -6 – 5x |NOTE: You are not separating the coefficient from the variable (the x |
|Get y by itself (move x). |-5x |from the 5), so you are not dividing both sides by 5. You are moving the|
| |0 |entire quantity of 5x, which is why you subtract. In other words, you are|
| |-3y = -5x -6 |moving the variable, not isolating it. |
|3. Solve so that y is positive and has a coefficient|-3 -3 -3 |NOTE: Write the sentence so that it follows the form. Therefore, you |
|of 1 (isolate y and make it positive). | |write -3y = -5x – 6, rather than -3y = -6 – 5x. |
| |y = 5/3x + 2 |NOTE: In this step you ARE separating the coefficient from the variable |
| | |(the -3 from the y), so you DO divide. In other words, you are isolating |
| | |the variable, not moving it. |
| | |NOTE: Write the slope (m) as a FRACTION, not a decimal. |
Example: Convert 10x – y = 6 to Slope-Intercept Form
|Write your original formula. | 10x – y = 6 – 10x |NOTE: You are not separating the coefficient from the variable (the x |
|Get y by itself (move x). |-10x |from the 5), so you are not dividing both sides by 10. You are moving |
| |0 |the entire quantity of 10x, which is why you subtract. In other words, |
| |(-1) -y = (-1)-10x + (-1)6 |you are moving the variable, not isolating it. |
|3. Solve so that y is positive and has a coefficient| |NOTE: Write the sentence so that it follows the form. Therefore, you |
|of 1 (isolate y and make it positive). |y = 10x + 6 |write -y = -10x – 6, rather than -y = -6 – 10x. |
2 Converting from Slope-Intercept Form to Standard Form
y = mx + b ( Ax + By = C
To be standard form:
1. No fractions anywhere (includes decimals): all whole numbers
2. Leading x cannot be negative
3. Must be in simplest form
4x + 2y = 10 ( 2x + y = 5
2 2 2
Example 1: Change y = 3/2x – 5 to Standard Form (Ax + By = C)
– 3/2x + y = (3/2x) – (3/2x) – 5
– 3/2x + y = -5 ( This is negative and a fraction, so it is not yet in standard form.
(-2)(-3/2)x + (-2)y = -5(-2)
-3x -2y = 10
Example 2: Change y = 1/5x – 1 to Standard Form (Ax + By = C)
– 1/5x + y = (1/5x) – (1/5x) – 1
– 1/5x + y = -1 ( This is negative and a fraction, so it is not yet in standard form.
(-5)(-1/5)x + (-5)y = -1(-5)
x -5y = 10
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Finding the Slope of a Line
Slope is rate of change
• It is the steepness of a line
• Represented by a fraction
m = slope = rise = ( y = change in y
run ( x change in x
m = 4 = 4
1
Positive slope
Negative slope
Two ways to find slope:
1. graph
2. formula m = y – y
x – x
Example: Find the slope of the line going through the points (-4, 4) and (4, 6)
| m = y – y | m = y – y |
|x – x |x – x |
| | |
|4 – 6 = -2 = 1 |6 - 4 = 2 = 1 |
|-4 – 4 -8 4 |4-(-4) 8 4 |
Example: Find the slope of the line going through the points (-4, -1) and (-2, -5)
m = y – y = -1 – (-5) = 4 = -2
x – x -4 – (-2) -2
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3/16/12
PARALLEL AND PERPENDICULAR LINES
Parallel Lines (||)
• Do not intersect
• Have the same slopes
Example: Write the slope-intercept form of an equation of a line parallel to y = 4x – 2 and passing through the point (-2, 2)
|1. Use the formula |y – y1 = m(x – x1) | |
|y – y1 = m(x – x1) | | |
|2. Organize your information | |m = 4 x1 = -2 y1 = 2 |
|3. Plug your values into your formula. |y – 2 = 4(x – (-2)) | |
|4. Solve |y – 2 = 4(x + 2) | |
| |y – 2 = 4x + 8 | |
| |+2 +2 | |
| |y = 4x + 10 | |
Perpendicular Lines (()
• Intersect
• Slopes are 1) opposite (change the sign)
2) reciprocal (flip them)
Example: Write the slope-intercept form of the ( line y = 1/2x + 1, crossing through point (4, 2).
|1. Use the formula |y – y1 = m(x – x1) | |
|y – y1 = m(x – x1) | | |
|2. Convert your slope | | Opp flip |
| | |m = ½ ( - ½ ( - 2/1 = -2 |
|3. Organize your information | |m = -2 x1 = 4 y1 = 2 |
|4. Plug your values into your formula. |y – 2 = -2(x – 4) | |
|5. Solve |y – 2 = -2x + 8 | |
| |+2 +2 | |
| |y = -2x + 10 | |
Example: Write the slope-intercept form of the ( line 2x + 4y = 12, crossing through point (-1, 3).
|1. Change the form of the original line from standard form | 2x + 4y = 12 - 2x | |
|to slope-intercept form. |-2x | |
| |4y = -2x – 12 | |
| |4 4 | |
| | | |
| |y = -1/2x - 3 | |
|1. Use the formula |y – y1 = m(x – x1) | |
|y – y1 = m(x – x1) | | |
|2. Convert your slope | | Opp flip |
| | |m = -½ ( ½ ( 2/1 = 2 |
|3. Organize your information | |m = 2 x1 = -1 y1 = 3 |
|4. Plug your values into your formula. |y – 3 = 2(x – (-1)) | |
|5. Solve |y – 3 = 2x + 2 | |
| |+3 +3 | |
| |y = 2x + 5 | |
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3/19/12
Line of Best Fit
Inside a SCATTERPLOT are data (ordered pairs). The more data you have, the better your line will be.
Why do we need a line? TO PREDICT! (for example, it helps business owners predict profits, whether to expand or downsize, etc).
|Line of Best Fit |( |Equation of a line to predict. |
|Regression Equation | | |
|Prediction Equation | | |
|Best Fit Line | | |
|X (L1) |Y (L2) |
|Independent Variable |Dependent Variable |
** See “Best-Fit Line” Handout**
Types of Questions you can be asked with Line of Best Fit
1. To find the equation
2. To make a prediction (from a line)
3. What does slope mean/represent?
4. What does y-intercept represent?
Example: The cab driver charges a $5 flat fee and $.25 per mile.
y = .25x + 5
y represents the total bill
.25 represents the charge or RATE
x represents per mile
5 represents the flat fee or starting point; the total bill starting at 0 miles
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4/9/12
Polynomial Operations
Monomial: one term
A number, a variable, or the product of 1 or more variables
Examples: 5, a, 6x, 5x2yz
Binomial: two terms
Addition or subtraction of two monomials
Examples: 5-5x; 2x2+3
Trinomial: three terms
Addition or subtraction of three monomials
Example: 6+7x2-3x
Adding Polynomials:
1) (4a - 5) + (3a + 6)
Combine like terms/regroup
(4a - 5) + (3a + 6)
4a + 3a - 5 + 6
7a + 1
2) (6xy + 2y + 6x) + (4xy – x)
10xy + 2y + 5x
Subtracting Polynomials:
|1) |(3a – 5) – (5a + 1) |
|DISTRIBUTE the -1 on the 2nd monomial |(3a – 5) + (- 5a – 1) |
|Combine like terms |- 2a - 6 |
|2) |(9xy + y – 2x) – (6xy – 2x) |
|DISTRIBUTE the -1 on the 2nd monomial |(9xy + y – 2x) +(- 6xy + 2x) |
|Combine like terms |3xy + y + 0 |
| |3xy + y |
| | |
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4/10/12
Multiplying Monomials
1. Multiply coefficient (whole number)
2. Add exponents
What does y3 mean? y * y * y
What does (x2)(x4) mean? x * x |* x * x * x * x ( x6
x2+4 = x6
EXAMPLES:
1) (x16) (x163) = x16+163 = x179
2) (4x3y2) (-2xy)
(4) (-2) x3+1 y2+1
-8 x4 y3
3) (-5xy) (4x2) (y4)
(-5) (4) (1) x3+1 y2+1
4) (-3j2k4) (2jk6)
(-3) (2) j2+1 k4+6
-6j3k10
Power to a Power
(x2)3 = x2 * x2 * x2 = x2+2+2 = x6
x x x x x x = x6
x2*3 = x6
RULE: Multiply exponents.
EXAMPLES:
1) (xy3)4 = xy3 * xy3 * xy3 * xy3 = x4 x12
= x1*4 y3*4 = x4 x12
4/11/12
Multiplying a Polynomial by a Monomial
RULE: Distribute!
EXAMPLES:
1) x(5x + x2)
x(5x) + x(x2)
5x2 + x3
2) -2xy(2xy + 4 x2)
(-2xy) (2y) + (-2xy) (4 x2)
-4x y2 + -8 x3y
3) 2x2y2(3xy + 2y + 5x)
6x3y3 + 4x2y3 + 10x3y2
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4/12/12
Binomials Times Binomials
There are 3 techniques you can use for multiplying polynomials:
1. Distributive Property
2. FOIL Method
3. The Box Method
EXAMPLE OF DISTRIBUTIVE PROPERTY OR FOIL METHOD:
(2x + 3) (5x + 8)
First: (2x)(5x) = 10x2
Outer: (2x)(8) = 16x
Inner: (3)(5x) = 15x
Last: (3)(8) = 24
Combine like terms: 10x2 + 31x + 24
EXAMPLES OF BOX METHOD:
(3x – 5) (5x + 2)
| |3x |-5 | |
|5x |15 x2 |-25x |15 x2 – 25x + 6x – 10 |
|2 |6x |-10 |15 x2 – 19x – 10 |
(2x – 5) (x2 – 5x + 4)
| |x2 |-5x |4 |2x3 - 10x2 - 5x2 + 8x + 25x - 20 |
|2x |2 x3 |-10 x2 |8x |2x3 - 15x2 + 33x - 20 |
|-5 |-5x2 |25x |-20 | |
4/13/12
Dividing Monomials
When dividing monomials, subtract the exponents
1. b5 = b * b * b * b * b = b5-2 = b3
b2 b * b
end
-----------------------
+ ? =
)
(
( find them by using the “Thumb Rule”
( This means x is 0, so this ordered pair will be (0, )
X
( This means y is 0, so this ordered pair will be ( , 0 )
X
( This means x is 0, so this ordered pair will be (0, )
X
( This means y is 0, so this ordered pair will be ( , 0 )
X
( This means x is 0, so this ordered pair will be (0, )
X
X
( This means y is 0, so this ordered pair will be ( , 0 )
Zero slope (m = 0)
negative
positive
Undefined (no slope)
................
................
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