methods
representations
solving
definitions
practice
100
The number or numbers that when substituted into an equation or inequality make the equation or inequality true. For example, x = 4 is a ______to the equation 3x − 2 = 10 because 3x − 2 equals 10 when x = 4. A _____ to a two-variable equation is sometimes written as an ordered pair (x, y). For example, x = 3 and y = −2 is a ______ to the equation y = x − 5; this ______can be written as (3, −2).
What is solution
100
The ____ used in this course represent numerical information by organizing it into columns and rows. The numbers may come from a graph, situation (pattern), or rule (equation). Many of the tables in this course are x-y tables like the one shown below.
What is table
100
Two or more straight lines on a flat surface that do not intersect (no matter how far they are extended) are parallel. If two lines have the same slope and do not coincide, they are ______. For example, the graphs of y = 2x + 3 and y = 2x − 2 are ______ (see diagram below). When two equations have _________ graphs, the equations have no solutions in common.
What is parallel
100
1. The ________ for a linear equation is ax + by = c, where a, b, and c are real numbers and a and b are not both zero. For example, the equation 2.5x − 3y = 12 is in _________. When you are given the equation of a line in ________, it is often useful to write an equivalent equation in y = mx + b form to find the line’s slope and y-intercept. 2. A quadratic expression in the form ax² + bx + c is said to be in ________. For example, the following are all expressions in ________: 3m² + m − 1, x² − 9, and 3x² + 5x.
What is standard form
100
y = 3x + 7 y = −4x + 21 coincide...
What is (2,13)
200
A method for solving a system of equations. The key step in using the ________ is to add or subtract both sides of two equations to eliminate one of the variables. For example, the two equations in the system below can be added together to get the simplified result 7x = 14. We can solve this equation to find x, then substitute the x-value back into either of the original equations to find the value of y.
What is elimination method
200
The Meaning of Slope and y-Intercept. in the Context of Word Problems. In the equation of a straight line (when the equation is written as "_______"), the slope is the number "m" that is multiplied on the x, and "b" is the y-intercept, where the line crosses the y-axis.
What is y = mx + b
200
This means that no matter what value is plugged in for the variable, you will ALWAYS get a contradiction
What is no solution
200
A mathematical summary (often an equation) of a trend in data, after making assumptions and approximations to simplify a complicated situation. ______s allow us to describe data to others, compare data more easily to other data, and allow us to make predictions. For example, mathematical _______sof weather patterns allow us to predict the weather. No ______is perfect, but some ______ are better at describing trends than other ______ . Regressions are a type of ______. Also see regression.
What is model
200
Solve for: 3x − y = 17 −x + y = −7
What is (5,-2)
300
A method for solving a system of equations. To use the _______, take two expressions that are each equal to the same variable and set those expressions equal to each other. For example, in the system of equations at right, −2x + 5 and x − 1 each equal y. So we write −2x + 5 = x − 1, then solve that equation to find x. Once we have x, we substitute that value back into either of the original equations to find the value of y.
What is equal values method
300
A ______represents numerical information spatially. The numbers may come from a table, situation (pattern), or rule (equation or inequality). Most of the graphs in this course show points, lines, and/or curves on a two-dimensional coordinate system like the one below or on a single axis called a number line (see below). Also see complete graph.
What is graph
300
A_________ is a point that the graphs of two equations have in common. For example, (3, 4) is a point of intersection of the two graphs shown below. Two graphs may have one _________ , several _________, or no __________. The ordered pair representing a _________ gives a solution to the equations of each of the graphs.
What is point of intersection
300
A ___________ is an equation that uses variables to represent unknown quantities. For example, the _______ b + g = 23 might represent the fact that the total number of boys and girls in the class is 23. It is helpful to define variables using “let” statements before using them in a __________ . Also see let statement.
What is mathematical sentence
300
Solve for... x = 3y − 52 x + 12y = −4
What is (-4,0.33)
400
Two graphs _______ if they have all their points in common. For example, the graphs of y = 2x + 4 and 3y = 6x + 12 coincide; both graphs are lines with a slope of 2 and a y-intercept of 4. When the graphs of two equations coincide, those equations share all the same solutions and have an infinite number of intersection points.
What is coincide
400
In the previous problem, you wrote equations that were models of a real-life ________.
What is situation
400
A_________ is a set of equations with the same variables. Solving a ________ means finding one or more solutions that make each of the equations in the system true. A solution to a ________ gives a point of intersection of the graphs of the equations in the system. There may be zero, one, or several solutions to a system of equations. For example, (1.5, −3) is a solution to the ________ below; setting x = 1.5, y = −3 makes both of the equations true. Also, (1.5, −3) is a point of intersection of the graphs of these two equations.
What is system of equations
400
A _______ is written at the beginning of our work to identify the variable that will represent a certain quantity. For example, in solving a problem about grilled cheese sandwiches, we might begin by writing “Let s = the number of sandwiches eaten.” It is particularly important to use ______ when writing mathematical sentences, so that your readers will know what the variables in the sentences represent.
What is let statement
400
Bob climbed down a ladder from his roof, while Roy climbed up another ladder next to him. Each ladder had 30 rungs. Their friend Jill recorded the following information about Bob and Roy: Bob went down 2 rungs every second. Roy went up 1 rung every second. At some point, Bob and Roy were at the same height. Which rung were they on?
What is (10,10)
500
A method for solving a system of equations by replacing one variable with an expression involving the remaining variable(s). For example, in the system of equations at right the first equation tells you that y is equal to −3x + 5. We can substitute −3x + 5 in for y in the second equation to get 2(−3x + 5) + 10x = 18, then solve this equation to find x. Once we have x, we substitute that value back into either of the original equations to find the value of y.
What is substitution method
500
A mathematical sentence in which two expressions appear on either side of an “equals” sign (=), stating that the two expressions are equivalent. For example, the equation 7x + 4.2 = −8 states that the expression 7x + 4.2 has the value –8. In this course, an ______ is often used to represent a rule relating two quantities. For example, a rule for finding the area y of a tile pattern with figure number x might be written y = 4x − 3.
What is equation
500
______ would mean that any value for the variable would make the equation, or system of equations true.
What is infinite solutions
500
the part of mathematics in which letters and other general symbols are used to represent numbers and quantities in formulae and equations.
What is algebra
500
Solve for x: 6x − 11 = 4x + 12
What is x=11.
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