Linear Functions & Systems By: Christian Knox and Shannon Gibley

Vocab System of Equations: 2 or more equations having 2 or more variables Solution set: a set of values that make every equation in the system true Absolute Value - The distance a number is from zero Inequality: the relation between two expressions that are not equal, employing a sign such as ≠ “not equal to,” > “greater than,” or < “less than.” Linear Equation: an equation between two variables that gives a straight line when plotted on a graph.

Linear Equations and Inequalities - Where do I begin? 2y = 16 Simplify what you already have Distribute and combine like terms on the same side. 2y + 10 = 16 If there are like terms on opposite sides of the equation, add/subtract them. 2y = ( ) = 6 Multiply and divide by the coefficient if needed. y = 3

Graphing Linear Equations + Inequalities Linear Equations Start with slope intercept form, “y = mx + b”. Place the y-intercept (b) on the axis. Use the slope to plot points on the graph, following the format “rise/run”. Linear Inequality Plot points as if it were a linear equation, but do not connect them with a line. Check the inequality sign. If it is > or <, draw a dotted line. If it is ≥ or ≤, connect them with a solid line. Shade above (> ≥) or below (< ≤) the line.

Absolute Value Equations & Inequalities Steps on how to solve an absolute value equation/inequality: If an equation- First isolate the absolute value onto one side Then, write 2 equations making on one equal a negative Lastly, solve and graph on a number line If an inequality- First, isolate the absolute value onto one side of the inequality Then, write 2 inequalities making one equal the opposite sign Lastly, solve and graph on a number

Systems of Equations: Substitution Method 2x - 3y = -2 4x + y = 24 If there is an equation with a variable on its own already, work from there. Solve for y. y = x Plug y in the first equation.

Systems of Equations: Substitution (cont.) 2x - 3(24 - 4x) = -2 Solve. x = 5 Plug the x value in for the y value. Y = (5) Y = 5

Systems of Equations: Elimination 2x - 3y = -2 3(4x + y = 24) First, you must line up the like terms vertically 2x - 3y = -2 12x + 3y = 72 14x=70 Add the vertically for a variable to cancel out ( for this step you might have to multiply one or both equations by a number to achieve this step )

Systems of Equations: Elimination(cont.) 14x=70 Solve for the remaining variable x= 5 Then, substitute the variable into one of the other equations to find the other variable 2(5)-3y= -2 Solve y=4

Solving system of equations(Word Problems) To solve a word problem you must: first read over your problem and determine what it wants you to do. The determine your X and Y’s in the equation Write the 2 equations needed and use the system of solving of your choice, unless provided, to solve for x or y Then take value of variable and plug back into equation to get 2nd value Finally, interpret the solutions

Practice Problems 1.4y + 5x = 65, y + 4x = 30. Solve for x and y. 2.2x+10x-29 > 5 solve for x 3.| 2x+10| - 12 = 60 solve for x 4.2x+10y=12 x+ 12y= 20 solve for x and y using Elimination Method 5. Lisa walks into a store and wants to buy 5 shirts and 4 pants. She in total has 9 items and it all together cost 40 dollars. Find out how much it cost per shirt and per pants.

Practice Problems (cont.) 6. 7x (x + 9) = 59. Solve for x. 7. The difference between two numbers is 3. Twice the smaller one added to four times the larger one is 48. What are the numbers? 8. y > 2x-5, Graph 9. 5|x + 10| = x + 6y = 1, 2x - 3y = 32. Solve for x and y.

Practice Problem Answers 6. X = and 9 9. x = -8, x = 13, y = Y = 10, x = 5 2.X > 5 3.x= 19, x= -4, y= 2 5.Shirts are 4 dollars, Pants are 5 dollars 8.