Introduction: Systems of Equations

Introduction: Systems of Equations

By Ashley Griffin

Multiple Choice

Warm Up!

After converting from standard form to slope-intercept form, what will the following equation look like?

$$4x+2y=6$$  

• $$y=2x+3$$  

• $$y=-\frac{1}{2}x-3$$  

• $$y=-2x+3$$  

• $$y=-\frac{1}{2}x+3$$  

More money, more problems

​You recently received some birthday money, $130 to be exact. You count the number of bills you have and find that you have 17 bills, all either $10 or $5 bills.

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How many of each bill do you have?​

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Hint: Just try guessing and checking first!​

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Poll

Which do you think is correct?

• nine $5 bills and eight $10 bills

• eight $5 bills and nine $10 bills

• ten $10 bills and seven $5 bills

• six $10 bills and eleven $5 bills

How does this have to do with math?

Sometimes, in math, we are asked for two answers, like how many $5 bills AND $10 bills. Systems of equations are how we solve these problems.

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You've lost me...

Systems of equations are two or more equations that have the same variables like:​

y=x+1

y=-2x+6​

which both have x and y​

Systems of Equations

Systems of equations are used when we are given two pieces of information and asked for two answers from it, like our first question.

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Solutions are where two lines intersect on a graph and these are our answers.

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The solution here is (-1,3) because that is where the lines intersect!​

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​Solutions=answers

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Multiple Select

What is a solution to a system of linear equations?

• The point where both lines intersect

• The point where the y-axis and x-axis meet

• An ordered pair that both lines share

• A salt dissolved in water

Multiple Choice

What would the solution be to this system of linear equations?

• (0,-1.5)

• (1, -3)

• (5,0)

• (0,0)

Multiple Choice

What would the solution be to this system of linear equations?

• (0,0)

• (2,2)

• (6,0)

• (0,4)

Multiple Choice

When would we get "infinitely many solutions" to a system of equations?

• The lines are parallel, never intersecting

• The lines cross at one point

• The lines are the same and overlap at all points

• The lines cross at 2 points

Multiple Choice

When will we get "no solutions" from a system of equations?

• The two lines cross once

• The two lines are the same and overlap

• The two lines are parallel and never touch

• The two lines cross at 2 points

Match

Match the following graphs to the systems of equations

 

 

  

   

• $$y=\frac{2}{3}x+1$$  

  $$y=\frac{2}{3}x-2$$  

• $$y=-x+7$$   $$y=2x+1$$  

• $$y=-2x+4$$   $$y=-2x+4$$   

Multiple Choice

Which of the following graphs would fit the system of equations?

$$y=2x-4$$  

$$y=-\frac{1}{2}x+1$$