Section 4.1 Solving Systems of Equations Integrated Math Section 4.1 Solving Systems of Equations

System of Equations- two or more equations Solution of a system- an ordered pair that satisfies the equations in the system.

Checking an ordered pair in a system Check the ordered pair in the equations in the system If the ordered pair checks in all the equations, it is a solution to the system.

Is the point (−6,8) a solution to the system. 2𝑥+𝑦=−4 𝑥−𝑦=−14 Is the point (−6,8) a solution to the system? 2𝑥+𝑦=−4 𝑥−𝑦=−14 *Plug in x and y.

Is (14, −16) a solution to the system. 2𝑥+4𝑦=−36 −3𝑥−6𝑦=54 Is (14, −16) a solution to the system? 2𝑥+4𝑦=−36 −3𝑥−6𝑦=54 *Plug in x and y

Try these! #1 Is (12,19) a solution to the system 𝑦=2𝑥−5 𝑎𝑛𝑑 𝑦= 3 5 𝑥−1 #2 Is 6,−3 a solution to the system 3𝑥+4𝑦=0 𝑎𝑛𝑑 7𝑥+27=36

There are several ways to find a solution to a system of equations. Method #1 Graph both lines on graph paper and find the point where the lines meet. Name the coordinates of the point of intersection. This is the solution to the system of equations

Change equations if necessary so they are in slope-intercept form. Graph each equation neatly on the same set of axes.

Graph the system of equations and find the solution. #1 𝑦=4𝑥−2 𝑎𝑛𝑑 2𝑥+𝑦=4 #2 𝑦= 2 5 𝑥+2 𝑎𝑛𝑑 𝑥+3𝑦=6

There are several ways to find a solution to a system of equations. Method #2 Graph the lines on the calculator and find the point of intersection .

Steps on the calculator Solve each equation for y Enter 1st equation in 𝑦 1 Enter 2nd equation in 𝑦 2 Make sure point of intersection is in the window. Adjust window if necessary. Press 2nd , trace, arrow down to #5 intersect, enter

A blinking spider will show up on the graph-use the left and right arrow to move the spider.

Spider on the left of the intersection point-enter Spider on the right of the intersection point-enter Enter, enter →𝑝𝑜𝑖𝑛𝑡 𝑜𝑓 𝑖𝑛𝑡𝑒𝑟𝑠𝑒𝑐𝑡𝑖𝑜𝑛

Find the point of intersection on the calculator! #1 𝑦=−2𝑥+10 𝑎𝑛𝑑 𝑦=2𝑥+2 #2 4𝑥+8𝑦=−48 𝑎𝑛𝑑 𝑥−3𝑦=28 #3 𝑦= 2 5 𝑥+2 𝑎𝑛𝑑 𝑥+3𝑦=6

Assignment #16A Pg. 231 #1-25 odd

Method #3 The substitution method

Finding x and y in the ordered pair #1 Use algebra to get (x) or (y) by itself in one equation. #2 Box in what it is equal to. #3 Put the “box” in the other equation for the variable you solved for. #4 Solve for the variable in the new equation. #5 Put the answer in the box to find the other variable. #6 Write the ordered pair.

Write the two numbers as an ordered pair Write the two numbers as an ordered pair. This is a solution to the system. #1 Solve the system of equations. 𝑥=𝑦+3 3𝑥−2𝑦=4

Solve the system of equations. 𝑥=𝑦+3 3𝑥−2𝑦=4 3(𝑦+3)−2𝑦=4 Solve for y! ( ,-5) Now find x! Go to the box and plug in -5. x=-5+3 Solution (-2,-5)

#2 Solve the system of equations #2 Solve the system of equations. 3𝑥+𝑦=2 −𝑥−3𝑦=6 Which equation would be easiest to solve for a variable?

Find the solution to the system! 2𝑥+3𝑦=33 𝑎𝑛𝑑 4𝑥−𝑦=17

Three different things can happen with a system of two equations #1 Parallel lines will never intersect and yield no common solution. If you are solving by substitution, both variables will disappear and the statement will be false.

Find the solution to the system 2𝑦−3𝑥=10 𝑎𝑛𝑑 4𝑦=6𝑥−4

#2 Equivalent equations will yield an infinite number of solutions-any point that satisfies the equation. If you are solving by substitution, both variables will disappear and you will have a true statement.

Solve the system of equations 2𝑥+3𝑦=15 𝑎𝑛𝑑 𝑦=− 2 3 𝑥+5

#3 The two lines will intersect in a point and the point will be the solution.

#1 Find the solution to the system 𝑦= 7 9 𝑥+2 𝑎𝑛𝑑 𝑦= 7 9 𝑥−6 #2 Find the solution to the system 4𝑥+2𝑦=10 𝑎𝑛𝑑 𝑦=−2𝑥+5 #3 Find the solution to the system 3𝑥−5𝑦=−14 𝑎𝑛𝑑 𝑦=2𝑥 #4 Find the solution to the system 𝑥+7𝑦=−20 𝑎𝑛𝑑 3𝑥+4𝑦=−9

Assignment #16B Pg. 232 #39-60 (x3)

Word problem steps #1 Identify the two variables-you can use any letters you want #2 Write two equations #3 Solve for the solution for the two variables using any method #4 Answer the question and label!

Look at pg 233 #82, 86, 90

Assignment #16C Pg. 233 #81-93 odd