1 / 18

ACC Module #2 Unit 2.5

ACC Module #2 Unit 2.5

pwkellysr
Télécharger la présentation

ACC Module #2 Unit 2.5

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. DemingEarly College High SchoolUnit 2.0 Quantitative Reasoning, Algebra, and Statistics (QAS) 2.5 Linear Equations

  2. Unit 2.0 QAS 2.0 Quantitative Reasoning, Algebra, and Statistics (QAS) 2.5 Linear Equations A function is called linear if it takes the form of the equation f(x) = ax +b or y = ax + b, for any two numbers a and b. A linear equation forms a straight line when graphed onto the coordinate plane.

  3. Unit 2.0 QAS 2.0 Quantitative Reasoning, Algebra, and Statistics (QAS) 2.5 Linear Equations A table of values and graph that satisfy the function:

  4. Unit 2.0 QAS 2.0 Quantitative Reasoning, Algebra, and Statistics (QAS) 2.5 Linear Equations When graphing a linear function, note that the ratio of change of the y coordinate to the change in the x coordinate is constant between any two points on the resulting line, no matter which two points are chosen. In other words, in a pair of points on a line, ,with so that the two points are distinct, then the ratio will be the same regardless of which particular pair of points are chosen.This ratio, , is called the slope of the line and is frequently denoted with the letter m and is equal to the “rise” over “run” of the line.

  5. Unit 2.0 QAS 2.0 Quantitative Reasoning, Algebra, and Statistics (QAS) 2.5 Linear Equations If the slope, m, is positive, then the lines upward when moving to the right.If the slope, m, is negative, then the line goes downward while moving to the right,If the slope, m, is 0, the line is called horizontal, and the y coordinate is constant along the entire line.In lines where the x coordinate along the entire line, y is not a function of x. For such lines, the slope is not defined. These lines are called vertical lines.

  6. Unit 2.0 QAS 2.0 Quantitative Reasoning, Algebra, and Statistics (QAS) 2.5 Linear Equations The standard equation for a linear equation (straight line) is given in the form:y = mx +bWhere y = f(x); m = slope and b = y-intercept. This is also called the “slope-intercept form”of the linear equation. Linear functions may take other forms than y = mx + b. The most common forms of linear equations are:Standard Form: Ax + By = C, in which the slope is given by , and the y-intercept is given by .

  7. Unit 2.0 QAS 2.0 Quantitative Reasoning, Algebra, and Statistics (QAS) 2.5 Linear Equations Slope-Intercept Form: y = mx +b. Where the slope is m and the y-intercept is b.Point-slope Form: where the slope is m and is any point on the chosen line.Two-Point Form: , where are any two distinct points on the chosen line. Note that the slope is: or “ ”.

  8. Unit 2.0 QAS 2.0 Quantitative Reasoning, Algebra, and Statistics (QAS) 2.5.1 Forms of Linear Equations Intercept Form: , in which is the x-intercept and is the y-intercept.These five ways to write linear equations are all useful in different circumstances. Depending on the given information, it might be easier to write one of these forms over another.If y = mx, y is directly proportional to x. In this case, changing x by a factor changes y by the same factor.If , y is inversely proportional to x. For example, if x is increased by a factor of three, then y will be decreased by the same factor, 3.

  9. Unit 2.0 QAS 2.0 Quantitative Reasoning, Algebra, and Statistics (QAS) 2.5.2 Solving Linear Equations Sometimes, rather than a situation where there’s an equation such as and finding y for some value of x is requested, the result is given and finding x is requested.The key to solving any equation is to remember that from one true equation, another true equation can be found by adding, subtracting, multiplying, or dividing both sides by the same quantity.In this case, it is necessary to manipulate the equation so that one side only contains the x terms. Then the other side will show what x is equal to.

  10. Unit 2.0 QAS 2.0 Quantitative Reasoning, Algebra, and Statistics (QAS) 2.5.2 Solving Linear Equations For example, in solving , adding 5 to each side results in . Next, dividing both sides by 3 results in . To ensure the value of x is correct, the number can be substituted into the original equation to see if the answer makes sense.In this case, can be simplified by cancelling out the 3’s. This yields 7 - 5 = 2. Which is a true statement.

  11. Unit 2.0 QAS 2.0 Quantitative Reasoning, Algebra, and Statistics (QAS) 2.5.2 Solving Linear Equations Sometimes an equation may have more than one x term. For example, consider the following equation:Moving all of the x terms to one side by subtracting x from both sides results in:Next, subtract 2 from both sides so that there is no constant term on the left side. This yields . Finally divide both sides by 2, which leaves:

  12. Unit 2.0 QAS 2.0 Quantitative Reasoning, Algebra, and Statistics (QAS) 2.5.3 Systems of Equations A system of equations is a group of equations that have the same variables or unknowns. These equations can be linear, but are not always so. Finding a solution to a system of equations means finding the values of the variables that satisfy every equation in the system.For a linear system of two equations and two variables, there could be a single solution, no solution, or infinite solutions.A single solution occurs when there is one value for x and y that satisfies the system. This would be shown on the graph where the lines cross at exactly one point. A good example of that are perpendicular lines.

  13. Unit 2.0 QAS 2.0 Quantitative Reasoning, Algebra, and Statistics (QAS) 2.5.3 Systems of Equations One Solution: solve and . Solution: Add these two equations. x + 3y = -3x - 3y = -3 2x = -6 →→ x = -3 Plug this into either equation - 3 + 3y = - 3 →→ y = 0 Solution: (-3,0)No Solution: solve, .Solution: There are no solutions because both of these lines have the exact same slope and never cross (they are parallel).If we try to solve this system of equations by adding or subtracting or by substitution we eliminate both variables.

  14. Unit 2.0 QAS 2.0 Quantitative Reasoning, Algebra, and Statistics (QAS) 2.5.3 Systems of Equations With infinitely many solutions, the equations may look different, but in fact, they are the same line. One equation will be a multiple of the other, and on a graph, they lie on top of each other.For example, solve Solution: These are actually the same line. If we factor out 3 from the second equation we get , the same as the first equation.The process of elimination can be used to solve a system of equations. For example, make up a system. Immediately adding or subtracting these equations does not eliminate a variable, but it is possible to change the first equation by multiplying it by -2. This changes the first equation to . These equations can then be added to get or y = 1.

  15. Unit 2.0 QAS 2.0 Quantitative Reasoning, Algebra, and Statistics (QAS) 2.5.3 Systems of Equations Solving for y yields y = 1. To find the rest of the solution(s), 1 can be substituted in for y in either of the original equations to find the value of x = 7. The solution to this system is (7, 1) because it makes both equations true, and it is the point in which the lines intersect.If the system is independent - having infinitely many solutions - then both variables will cancel out when the elimination method is used, resulting in an equation that is true for many values of x and y.Since the system is dependent, both equations can be simplified to the same equation or line.

  16. Unit 2.0 QAS 2.0 Quantitative Reasoning, Algebra, and Statistics (QAS) 2.5.3 Systems of Equations A system can also be solved using substitution. This involves solving one of the equations for a variable and then plugging that solution into the other equation in the system. For example, can be solved by using substitution. The first equation can be solved for x, where . Then it can be plugged into the other equation: , solving for y yields That yields y = 3. If y = 3, then x = 1.This solution can be checked by plugging in these values for the variables in each equation to see if it makes a true statement.

  17. Unit 2.0 QAS 2.0 Quantitative Reasoning, Algebra, and Statistics (QAS) 2.5.3 Systems of Equations Finally, a solution to a system of equations can be found graphically. The solution to a linear system is the point or points where the lines cross. The values of x and y represent the coordinates (x, y) where the lines intersect.Using the same system of equations as in the previous problem, they can be solved for y to put them in the slope-intercept form, . These equations become . The slope is the coefficient of x, and the y-intercept is the constant value (i.e. -4.5).

  18. Unit 2.0 QAS 2.0 Quantitative Reasoning, Algebra, and Statistics (QAS) 2.5.3 Systems of Equations A system of equations may also be made up of a linear equation and a quadratic equation. These systems may have one solution, two solutions, or no solutions.The graph of this type system involves one straight line and one parabola.Algebraically, these systems can be solved by solving the linear equation for one variable and plugging that solution into the quadratic equation.If possible, the equation can then be solved to find part of the answer. The graphing method is commonly used for these types of systems. On a graph, these two lines can be found to intersect at one point, two points, or at no points.

More Related