Solving systems of equations through given points and graphing Systems and Inequalities A solution to a system of equations is an ordered pair of numbers that satisfies both equations in a system. Systems of Linear Equations Determine whether the given point is a solution to the system of equations. For Example: x + y = 10

3x - 4y = 9 a) (7,3)

b) (5,5) a) 7 + 3 = 10

3(7) - 4(3) = 9

21 - 12 = 9

This ordered pair is a solution for this system of equations. b) 5 + 5 = 10

3(5) - 4(5)

15 - 20 = -5

This ordered pair is not a solution to this system of equations Check whether each ordered pair is a solution of the systems of equations. Now try a few! 1)

x + y = 7

2x - 7y = -31

a) (3,4)

b) (2,5) 2)

xy = 12

5y + 3x = 28

a) (6,2)

b) (4,3) One way to solve a system of two equations is to graph both equations and identify any points of intersection. The coordinates of each intersection represent a solution to that system. Solving a System of Equations How to Solve a System of Equations by Graphing Solve the system:

x + y = 10

x - y = 15 Write both equations in the slope-intercept form (y = mx + b). Step 1 x + y = 10

-x -x

y = -1x + 10

x - y = 2

-x -x

-1y = -1x + 2

Divide -1 from both sides

y = x - 2 Graph the first equation. Step 2 Plot b Then count m up or down

the slope is like a fraction, so because the slope of this equation is -1, you would write it as -1/1 and move the point down one and over (to the right) one. Graph the second equation. Step 3 The answer is the point of intersection. Write this in the ordered pair form and check the solution. Step 4 (6,4) x + y = 10

6 + 4 = 10 x - y = 2

6 - 4 = 2 This pair is the solution to this system of equations. Solve the following systems of equations by graphing and state the solution. Try it out! 3)

x + y = 20

2x - y = 13 4)

x + y = 12

2x - 3y = -6 5)

x + y = 23

1/2x - y = -2 These equations contain x and y which mean they fit the

y = mx + b model. Don't panic about the inequality sign. This only means that instead of solutions lying solely on the line, they can lie in the indicated direction. Graphing Linear Inequalities Determine whether the given point is a solution of 5x - 4y > 13 For Example: a) (-3,2)

b) (3,0) 5(-3) - 4(2) > 13

-15 - 8 is not greater than 13, therefore the point is not a solution. 5(3) - 4(0) > 13

15 - 0 > 13

This point is a solution because 15 is greater than 13. The solution set for a linear inequality in two variables is always a half-plane with either a dotted or solid border. In layman's terms, the line created by the inequality splits the plane in half. Here's where it gets a bit complicated... When you see > or <, draw a dotted line. When to draw what kind of line When you see a less than or equal to sign OR a greater than or equal to sign, draw a solid line. The one-point test is a way to determine which side of the line to shade in a two-variable inequality. The One-Point Test To do this, use the y = mx + b form to sketch the line (replacing the = with an inequality sign). Then choose a point (typically 0,0 if it is not already on the line) to determine if it is true or false. If the inequality is true, shade the piece of the graph that contains the point. If the inequality is false, shade the piece of the graph that does not contain the point. 6) 8x + 3y (greater than or equal to) 12 Test what you've learned! If a negative number is divided across the inequality, the sign switches.

For example:

6x - 2y < 12

subtract 6x from both sides

-2y < -6x + 12

now divide by -2 to get y by itself

y > 3x - 6 One last thing... 8) x - 3y (greater than or equal to) 9 9) 5x - 2y < 0 7) 2(x + y) > 24 Psh. Too fast. Go back and check your work! Finished already? 1) b Let's see how you did! 2) b 3) (11,9) 4) (6,6) 5) (14,9) 6) False 7) False 8) False 9) True Woohoot! Congrats! You've finished!

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# Systems and Inequalities

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