Introduction: Systems of Equations

Introduction: Systems of Equations

By Ashley Griffin

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

Fill in the Blank

The first step of solving a system of linear equations is putting the equations in ______-intercept form

Fill in the Blank

The second step to solving a system of linear equations is to graph the ____-intercept and use slope to create the line. We do this for both equations.

Fill in the Blank

The third step is to identify the point where the lines intersect aka the ________

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?

$$y=2x-4$$  

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