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Mr. Chadalavada(615)[email protected]:10:00a.m.–2:00p.m.
Geometry
LearningPacketOverviewThislearningpacketwillhavethreesectionswithmaterialrangingfromsimplifyingexpressionstographinglinearequations.Therearecondensednotesforyoutoreviewbeforeattemptingproblemsattheendofeachsection.Pleasereviewthispacketoverthenexttwoweeks.Iwillreleasetheanswerstotheproblemson3/30/2020.PleasesignupfortheGeometry-ChadalavadaGoogleclassroomwiththeclasscodelximdia.
NecessaryMaterials● Youmayusecalculators,notes,andanyresourcetoreviewthismaterial.
● Khanacademyisagoodtoolforreviewmaterialonanyspecifictopic
● Feelfreetoreachouttomeatanytimeonquestionsaboutthesetopics.
Part I: Simplifying Expressions and Combining Like Terms Order of Operations Review: When an expression contains more than one operation, the operations must be performed in a certain order.
I. Evaluate any expressions inside grouping symbols like ( ) or [ ] II. Evaluate exponents
III. Perform multiplication and division in order from left to right IV. Perform addition from left to right.
Many people remember this using either of the following acronyms:
x PEMDAS (Parenthesis, Exponents, Multiplication, Division, Addition Subtraction)
Some people use this saying to remember PEMDAS: Please Excuse My Dear Aunt Sally
x GEMS
(Grouping Symbols, Exponents, Multiplication & Division, Subtraction & Addition)
Examples:
The following symbols are also considered as grouping symbols when using the order of operations.
Evaluate each expression. |1| –
|2| –
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(
)
Evaluate each expression using the values given.
|9| use and
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use and
Simplifying Expression with Variables Review: Combining like terms:
x Terms may only be combined (added/subtracted) if they are like terms. Like terms may have different coefficients (the number multiplied in front of the variable) but must have all of the same variable(s) and all of the same exponents.
o Examples
In order to combine like terms, you might use some of the following properties:
x Associative Property of Addition/Multiplication: o If all terms share the same operation, the operation may be applied in any order with the same result. o Examples:
x Distributive Property: o If there is a single term multiplied (outside parenthesis) to an expression inside parenthesis, the term may be
distributed and multiplied separately to each term inside the parenthesis. o **Don’t forget that when a negative is outside of the parenthesis, it is equivalent to distributing a -1 to the
expression inside the parenthesis** o Examples:
Simplify:
Simplify: – – – – – –
Simplify:
o If there are expressions being multiplied that both have more than one term, you must be sure to distribute every
term in the first expression to every term in the second expression � When multiplying two binomials (expressions that each have two terms) many use the acronym F.O.I.L.
(first, outer, inner, last) to remember to multiply all terms.
o Examples:
Simplify:
Simplify:
Simplify each of the following expressions. |1| |2|
|3| |4|
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|9| |10| – –
|11| – |12| – –
|13| – – – – |14| – – –
|15| – |16| –
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Simplifying Exponential Expressions Exponent Rules:
To simplify expressions with exponents, you must follow these exponent rules: Rule: Rule: Example: Product Rule: when terms with the same base are multiplied, you add the exponents
Quotient Rule: When terms with the same base are divided, you subtract the exponents (numerator exponent – denominator exponent)
Power Rule: When a term is raised to another exponent, you multiply the exponents.
Power of a Product/Quotient Rule: When a group of variables being multiplied or divided is being raised to a power, you may distribute the exponent and use power rule for each variable.
(
)
(
)
Zero Exponent: Anything raised to the zero power is equal to one.
Negative Exponents: A negative exponent is equivalent to taking the reciprocal of the base of the exponent and applying the absolute value of the exponent.
( )
( )
( )
(
)
(
)
**Remember that the Order of Operations still applies here – parenthesis must always be taken care of first**
Simplify each of the following expressions. Your answers should include only positive exponents. |1|
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|5| |6| (
)
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Part II: Solving Linear Equations The process behind solving equations with one variable follows the ideas of inverse operations. To evaluate a numerical expression, you use order of operations (PEMDAS). To solve an equation for a variable, you must undo the order of operations (reverse PEMDAS or SADMEP) The rules for solving linear equations can be summarized as:
1. Distribute if necessary. 2. Combine like terms (Subtraction/Addition)
a. You might need to move terms to different sides of the equal sign in order to combine. i. We do this by doing opposite operations with the variables. For example, if a variable is added on one side,
you would subtract it to move it to the other side. If a variable was subtracted, you would need to add it to the other side.
3. Isolate the variable – get the variable alone on one side of the equal sign. a. Add/subtract to move numerical terms to the other side. b. Multiply/divide to remove coefficients. (Multiplication/Division)
Examples:
**Always check your answer by plugging the value into the original equation**
Solve each of the following equations. |1|
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Solving a Proportion A proportion is an equation that has fractions on both sides of the equal sign. The key to solving a proportion is in “clearing the denominators”. Simply, all you need to do is multiply both sides of the equation by each of the denominators, one at a time. Example:
Solve: n Work:
After this, you solve the resulting equation like all other equations.
Solve each of the following.
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Solving for a variable in a formula Solving for a single variable in a formula can make it easier to use that formula. This is often used with physics formulas to ensure easier calculations. The process is similar to that of solving multi-step equations: find the operations being performed on the variable you are solving for, and then use inverse operations to isolate it.
Solve for the indicated variable.
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Part III: Linear Equations and Graphing Graphing in the Coordinate Plane: The Coordinate Plane is a grid that is defined by a horizontal axis called the x-axis and a vertical axis called the y-axis. It is a way of graphing locations called coordinates, which look like (x, y), where x is the location on the x-axis and y is the location on the y-axis. When two coordinates (often called points) are connected, they form a line that has what is called slope. Slope is the measure of the “steepness” of a line, written as a fraction:
Given two points , you may find the slope of the line connecting them by graphing and counting the rise and run or by using the formula:
Examples:
Every line can be written using an equation, which provides a map for any of the points that lie on the line. There are many ways to write the equation of a line, but the most frequent is: Slope-Intercept Form – If an equation is written in slope-intercept form, the slope can be identified as the in the equation and the y-intercept (the point where the line intersects the y-axis) can be identified as . If an equation is in Slope-Intercept Form, the equation can easily be graphed in a coordinate plane by following the steps:
1. Identify the slope (m) and y-intercept (b) of the equation. 2. Plot the y-intercept on the graph 3. Count the slope (rise/run) away from your y-intercept to plot the next point, 4. Use the slope to plot a few more points for accuracy and connect the points to form a line.
If an equation is not written in slope-intercept form, you can always re-write the equation by solving for y. Example:
The equations for horizontal and vertical lines cannot be written in slope-intercept form. This is because their slopes do not have both a rise and a run. Instead, we use the following rules.
Vertical Lines: Horizontal Lines: Equation: x = 3
Equation: y = -2
This graph has infinite rise and no run. A vertical line’s
slope is called undefined. This graph has infinite run and no rise. A horizontal
line’s slope is zero. Always take the form:
x = # Always take the form:
y = #
Find the slope of the line that passes through each pair of points. Find the slope of the line shown in each grid. |1|
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Graph each of the following equations in the provided grid.
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Writing the equations of lines We can also use slope-intercept form to write the equations of lines by plugging in given information like the slope, y-intercept, and/or other points on the line. Examples:
Write the slope-intercept form of the equation of the line shown in each grid.
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Write the slope intercept form of the equation that satisfies each of the following requirements.
|3| Slope = -2 , y-intercept = 1 |4| Slope = -1 , y-intercept = -3
|5| Slope = , passes through the point
|6| Slope = , passes through the point
Write the following equations in slope-intercept form.
|7| |8| – |9|
slope
Parallel and Perpendicular Lines: Two lines are parallel if they never intersect. In order to never intersect, parallel lines must have the same slope.
Two lines are perpendicular if they intersect at a 90 angle. In order to intersect this way, perpendicular lines must have slopes that are reciprocals and the opposite signs of each other.
Examples:
slope same slopes
Reciprocal, opposite sign slopes
State whether each pair of lines is parallel, perpendicular, or neither. |1|
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