Study Guide 1, Midterm 1, Math 40

1. demand equation 2. supply equation 3. equilibrium price

1. The demand equation gives the number of units of the product that consumers will buy at a given price. 2. The supply equation gives the number of units that the producer will supply at that price. 3. The price at which the supply and demand are equal is called the equilibrium price. This is the price at which the consumer and the producer agree to do business.

Inconsistent and Dependent Systems When using elimination to solve a system:

Inconsistent and Dependent Systems When using elimination to solve a system: 1. If combining the two equations results in an equation of the form 0x + 0y = k (k ≠ 0) then the system is inconsistent. 2. If combining the two equations results in an equation of the form 0x + 0y = 0 then the system is dependent.

extrapolation

Making predictions beyond the range of known data is called extrapolation. Extrapolation can often give useful information, but if we try to extrapolate too far beyond our data, we may get unreasonable results.

Using a regression line to find an equation

Pick two points on the line, which may or may not be actual graph points. Mark those points with an X to identify them as different than the actual points. Then find the slope using the two points. (slope formula; m = y _2 − y_1 / x_2 − x_1) After finding the slope, use one of the points and the slope (point-slope; m = y − y_1 / x − x_1) to find an equation as you usually do.

2.1 Scatter-plot

The data on a graph are not strictly linear, because the slope is not constant. However, the data points do appear to lie close to an imaginary line. Draw a line that comes as close as possible to the data points. Adjust the line so there are about the same number of points above the line as there are below the line.

Graph

The graph of an equation is a picture of all the variables that make an equation true.

Positive slope on a graph or increasing graph Negative slope on a graph or decreasing graph

The line rises left to right The line falls left to right

ordered pair

The solution of an equation consists of an ordered pair of values, one for x and one for y, together they satisfy the equation (make it true). (x,y)

Solution

The solution to an equation: the value of the variable(s) make the equation true.

Slope

The steepness of a line that measures the rate of change.

x-intercept and y-intercept

The x-intercept is where the line on a graph crosses the x axis and the y-intercept is where a line on a graph crosses the y axis.

To Solve a System by Elimination:

To Solve a System by Elimination: 1. Write each equation in the form Ax + By = C. 2. Decide which variable to eliminate. Multiply each equation by an appropriate constant so that the coefficients of that variable are opposites. 3. Add the equations from Step 2 and solve for the remaining variable. 4. Substitute the value found in Step 3 into one of the original equations and solve for the other variable.

To Graph a Linear Equation by the Intercept Method

1. Find the horizontal and vertical intercepts. 2. Plot the intercepts, and draw the line through the two points. To find the intercepts you solve the equation by making the y = 0 to get the x-intercept (x, 0) where the line crosses the x axis ; then make the x = 0 and solve for y to get the y-intercept (0, y) where the line crosses the y axis.

General form of a linear equation (standard form)

Ax + By = C Remember: (where both A and B cannot be 0)

outliers

Data points that lie far from the regression line are called outliers

2.3 To Solve a System by Substitution:

To Solve a System by Substitution: 1. Choose one of the variables in one of the equations. (It is best to choose a variable whose coefficient is 1 or −1.) Solve the equation for that variable. 2. Substitute the result of Step 1 into the other equation. This gives an equation in one variable. 3. Solve the equation obtained in Step 2. This gives the solution value for one of the variables. 4. Substitute this value into the result of Step 1 to find the solution value of the other variable.

regression line

A line that fits the data in a scatterplot is called a regression line

Explain what ∆x and ∆y mean

∆x and ∆y mean the change in x and the change in y.

Linear equation

A linear equation is a straight line on a graph that represents all solutions for that equation.

Linear model

A linear model describes a variable that increases or decreases at a constant rate. It has the form y = (starting value) + (rate) x t

Rate

A rate is a ratio between two quantities. The 'RATE OF CHANGE' is a special kind of ratio that compares the change in two quantities or variables.

a regression line is a type of linear model used for...

A regression line is a type of linear model. We can use it to analyze data and to make predictions.

Mathematical Model

A simplified description of reality that helps us understand a system or process. A mathematical model can be a graph, equation, diagram, scatter-points that represent a real life situation. It helps us understand a system or process.

2.2 What is the solution of a linear system of two equations in two unknowns (or a 2 × 2 linear system, for short).

A solution to the system is an ordered pair (x, y) that satisfies both equations in the system. The solution is the point that the two lines cross on a graph. That point satisfies both equations.

1. consistent system-infinite solutions-is dependent-both equations are on the same line 2. dependent system-no solutions-inconsistent-equations are parallel on a graph

If a consistent system has an infinite number of solutions, it is DEPENDENT . When you graph the equations, both equations represent the same line. If a system has no solution, it is said to be INCONSISTENT . The graphs of the lines do not intersect, so the graphs are parallel and there is no solution.

Know how to use the slope-intercept form to find an equation.

If you are using the slope-intercept form to find an equation of a line you need to find the slope (use m = ∆y / ∆x). Now you have the slope and one point. se one point and the slope to solve for b. Now you have the needed slope and the y=intercept so you can write your equation in the form of y = mx + b.

Rise/run

Rise/run is the slope, m = ∆y / ∆x The slope = the change in y / the change in x

Solutions of Linear Systems There are three types of 2 × 2 linear systems:

Solutions of Linear Systems There are three types of 2 × 2 linear systems: 1. Consistent and independent system. The graphs of the two lines intersect in exactly one point. The system has exactly one solution. 2. Inconsistent system. The graphs of the equations are parallel lines and hence do not intersect. An inconsistent system has no solutions. 3. Dependent system. All the solutions of one equation are also solutions to the second equation, and hence are solutions of the system. The graphs of the two equations are the same line. A dependent system has infinitely many solutions.

Point slope form

Use this formula when one point and the slope are given.

Find the equation of a line given the slope and one point.

Use your given point (x,y) and your given slope and solve for b (the y-intercept). Now plug in your slope and y-intercept into y = mx + b and you have the formula for that line. You can graph the line using your y-intercept and the point given, draw your line on your graph.

interpolation

Using a regression line to estimate values between known data points is called interpolation. If the data points lie fairly close to the regression line, then interpo-lation will usually give a fairly accurate estimate

Coordinate formula for slope-explain and state

Using any two points on a line graph, using the formula that slope = change in y / change in x

scale/spaces on an axis

labeling the scale on the x and y axis with relevant numbers.

net change

net change = final value - starting value The net change is positive if the variable increases, and negative if it decreases.

graphing an inequality

on a number line if the point is equal to or greater than you fill in the circle. If the point is greater than or less then the point is an open circle.

Find the slope on a linear graph

rise / run The rise is the y axis and the run is the x axis

Explain slope intercept form

y = mx + b