Solving Linear Equations Portfolio
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Investigate and explain the different types of solutions to a system of equations.
Question 1
Examine the six different equations that follow. Determine which pairs of equations have one solution, no solution, and infinite solutions. On your worksheet,
put a circle around the two equations that have infinite solutions.
put a square around the two equations that have no solution.
underline the two equations that have one solution.
2x+3=2x+7 (circle)
7x+3=7x+3 (underline)
4x+22=8x+10(square)
5x+5=5x+4(circle)
3x+18=5x+8 (underline)
3(4x+8)=4(3x+6)(square)
What do you notice about equations that are in each of these categories? Use complete sentences to answer the questions that follow.
Question 2: Infinite Solutions
Consider the two equations you circled, which form a system with infinite solutions? Solve the equations. What do you notice about these equations? What does it mean to have infinite solutions? Plug more than one point into the original equations and check to show that the equations can have more than one solution. Explain your work.
Question 3: No Solutions
Consider the two equations you put a square around, which form a system with no solution. Solve the equations. What do you notice about these equations? What does it mean to have no solution? Plug a few points into the original equations and check to show that they can have no solution. Explain your work.
Question 4: One Solution
Consider the two equations that you underlined, which form a system with one solution. Solve the equations. What do you notice about these equations? What does it mean to have one solution? Plug a few points into the original equations and check to show that the system of equations has one solution. Explain your work.
Question 5: Writing Linear Equations
Now try writing and solving your own systems of linear equations: one with infinite solutions, one with no solution, and one with one solution. Write a real-world scenario that goes with at least one system of equations. Show your work.
Question 2: Infinite Solutions
The two equations 2x+3=2x+7 and 5x+5=5x+4 form a system with infinite solutions. When we solve these equations, we can simplify them to 0=4 and 0=4, which means they are both equal and there are infinitely many solutions. This means that any value of x will satisfy these equations.
For example, if we substitute x=3 into the original equations, we get:
2(3)+3 = 2(3)+7
6+3 = 6+7
9 = 9
And,
5(3)+5 = 5(3)+4
15+5 = 15+4
20 = 20
This shows that both equations are satisfied by x=3, as well as any other value of x.
Question 3: No Solutions
The two equations 4x+22=8x+10 and 3x+18=5x+8 form a system with no solution. When we solve these equations, we can simplify them to 0=12 and 0=10, which means they are contradictory and there are no values of x that satisfy both equations.
For example, if we substitute x=3 into the original equations, we get:
4(3)+22 = 8(3)+10
12+22 = 24+10
34 = 34 (first equation satisfied)
And,
3(3)+18 = 5(3)+8
9+18 = 15+8
27 = 23 (second equation not satisfied)
This shows that there is no value of x that satisfies both equations.
Question 4: One Solution
The two equations 7x+3=7x+3 and 3(4x+8)=4(3x+6) form a system with one solution. When we solve these equations, we can simplify them to 0=0 and 0=0, which means they are both identities and have infinite solutions. In this case, any value of x will satisfy the equations.
For example, if we substitute x=3 into the original equations, we get:
7(3)+3 = 7(3)+3
21+3 = 21+3
24 = 24 (first equation satisfied)
And,
3(4(3)+8) = 4(3(3)+6)
3(12+8) = 4(9+6)
3(20) = 4(15)
60 = 60 (second equation satisfied)
This shows that there is one value of x (in this case, x=3) that satisfies both equations.
Question 5: Writing Linear Equations
One with infinite solutions:
2x+4 = 2x+4
This system of equations has infinite solutions, as any value of x will satisfy both equations.
One with no solution:
3x+5 = 3x+8
This system of equations has no solution, as the two equations are contradictory.
One with one solution:
4x+6 = 2x+10
This system of equations has one solution, as there is one value of x that satisfies both equations.
Real-world scenario:
Let x represent the number of adults and y represent the number of children attending a concert.
Equation 1: 2x + 3y = 20 (The total number of attendees is 20)
Equation 2: x + y = 12 (The total number of people is 12 adults and children)
This system has one solution, where x=8 and y=4.