Section 7 – 1 Solving Systems by Graphing 2 or more linear equations make a system of linear equations The solution to a system of equations is the point where they all cross One way to solve systems of linear equations is to graph them and then see where they cross (this is only a good option when the solutions only contains integers)

Ex1. Solve by graphing

Graphing calculators can give you exact points 1) hit Y= and input the equations 2) hit GRAPH3) hit 2 nd – CALC 4) chose INTERSECT 5) use the → and ← to move the blinking light near the intersection and hit ENTER 6) blinking light jumps to other line, so hit ENTER again 7) use the arrows again to guess where they intersect and hit ENTER – answer is given

Ex2. Solve If the lines are parallel, there is no solution because they do NOT cross If the graphs of all equations are the same, then there are infinitely many solutions (every point on the line) Solve by graphing Ex3. Ex4.

Section 7-2 Solving Systems Using Substitution Another way of solving systems (and one that is more precise) is by substitution With substitution, you must solve for a variable in one of the equations and then substitute that value in for that variable in another equation Solve using substitution Ex1. Ex2.

Section 7-3 Solving Systems Using Elimination Another way to solve systems algebraically is by elimination 1) write both equations in standard form 2) multiply 1 or both equations so that the coefficients for 1 variable are opposites 3) add the columns 4) solve for the remaining variable 5) plug the answer into any equation to solve for the other variable

Solve by elimination. Ex1. Ex2. Ex3.

Section 7-4 Applications of Linear Systems In this section you will have to write two linear equations and then solve the system Now that you know 3 methods for solving, pick your favorite for this section (an algebraic method is best) Ex1. A chemist has one solution that is 50% acid. She has another that is 25% acid. How many liters of each type of acid solution should she combine to get 10 liters of a 40% acid solution?

Ex2. Suppose you have a typing service. You buy a computer for $1750 on which to do your typing. You charge $5.50 per page for typing. Expenses are $.50 per page for ink, paper, electricity, and other expenses. How many pages must you type to break even?

Section 7-5 Linear Inequalities The solutions of an inequality are all of the points that make the inequality true When graphing linear inequalities, you will have to shade on one side of the inequality (the side that makes the inequality true) are graphed with a dashed line are graphed with a solid line

To determine if a point is a solution to an inequality, plug in the coordinates for x and y into the inequality, if the statement is true, then it is a solution To determine where to shade (2 methods): A) put the inequality in slope-intercept form and if it is or > then you shade above B) pick a test point (not on the line) to determine if that point is a solution (if it is, shade on that side, if not, shade on the opposite side)

Ex1. Is (-3, 5) a solution to y < -2x + 4? When you graph inequalities, you can only put one question per graph Ex2. y > -⅜x + 5 Ex3. y < 2x – 4 Ex4. 2x – 3y > -12

Section 7-6 Systems of Linear Inequalities To solve a system of linear inequalities, graph each of the inequalities (shade very lightly) and the solution is the overlap of the shading Make sure it is still clear which lines are dashed vs. solid and where the shading is for your final answer (should be shaded darker or in a bolder color)

For a point to be a solution to a system of inequalities, it must be true for both inequalities Solve by graphing. Ex1. y 4 Read example 3 on page 379