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+37x+34x+225x+53x+183(4x+8)======2x+77x+38x+105x+45x+84(3x+6)
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.
Solving Linear Equations Portfolio Worksheet
Use the rubric to check your work on the portfolio.
Solving Linear Equations Portfolio Rubric
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Unfortunately, as an AI language model, I cannot read or access any external files or worksheets. However, I can provide you with guidance on how to approach each question in the portfolio.
Question 1: Examine the six different equations and determine which have one solution, no solution, and infinite solutions. Put a circle around equations with infinite solutions, put a square around equations with no solution, and underline equations with one solution.
To determine the type of solution for each equation, you need to solve the equations using appropriate methods like substitution or elimination. For equations with infinite solutions, there will be a relationship between the equations, and for equations with no solution, you will reach a contradiction. Equations with one solution will have a unique solution for both variables.
Question 2: Solve the two equations that form a system with infinite solutions. Notice any patterns or relationships between the equations, and explain what it means to have infinite solutions. Plug in more than one point into the original equations to demonstrate that there can be more than one solution.
When solving the equations with infinite solutions, you should end up with the same equation on both sides of the equation. This means that any value of the variables that satisfies one equation will also satisfy the other equation. By plugging in different values for the variables, you can demonstrate that there are infinitely many solutions.
Question 3: Solve the two equations that form a system with no solution. Notice any patterns or relationships between the equations, and explain what it means to have no solution. Plug in a few points into the original equations to demonstrate that they can have no solution.
When solving the equations with no solution, you will reach a contradiction or an inconsistent statement when simplifying the equations. This means that there is no combination of values for the variables that will satisfy both equations simultaneously. By plugging in various points, you can demonstrate that the equations are not satisfied by any particular combination of values.
Question 4: Solve the two equations that form a system with one solution. Notice any patterns or relationships between the equations, and explain what it means to have one solution. Plug in a few points into the original equations to demonstrate that the system has one solution.
When solving the equations with one solution, you will obtain a unique and specific solution for both variables. The equations will intersect at a single point on a graph, representing the solution to the system. By plugging in different points, you can demonstrate that the equations only satisfy a particular combination of values.
Question 5: Write and solve your own systems of linear equations. Create one with infinite solutions, one with no solution, and one with one solution. Provide a real-world scenario to explain the system of equations.
For each scenario, you will need to define the variables, write down the equations based on the given information, and solve the system using appropriate methods. Make sure to demonstrate the type of solution for each system (infinite solutions, no solution, or one solution) and explain how it relates to the real-world scenario you provided.
Remember to refer to the provided rubric to check your work and ensure you have addressed all the requirements for the portfolio.
Question 1:
From the given equations:
2x + 3
7x + 3
8x + 10
5x + 4
5x + 8
4(3x + 6)
Circle the equations that have infinite solutions:
2x + 3 = 7x + 3
5x + 4 = 5x + 8
Square the equations that have no solution:
8x + 10 = 5x + 4
4(3x + 6) = 5x + 8
Underline the equations that have one solution:
2x + 3 = 5x + 8
7x + 3 = 5x + 4
Question 2: Infinite Solutions
First equation: 2x + 3 = 7x + 3
Subtracting 2x from both sides:
3 = 5x + 3
Subtracting 3 from both sides:
0 = 5x
Dividing both sides by 5:
0/5 = x
x = 0
Second equation: 5x + 4 = 5x + 8
Subtracting 5x from both sides:
4 = 8
This equation is inconsistent, meaning there is no solution.
Having infinite solutions means that all points on the line satisfy the equation. In this case, the equations are identical and represent the same line. Therefore, any value of x will give a solution.
Question 3: No Solutions
First equation: 8x + 10 = 5x + 4
Subtracting 5x from both sides:
3x + 10 = 4
Subtracting 10 from both sides:
3x = -6
Dividing both sides by 3:
x = -2
Second equation: 4(3x + 6) = 5x + 8
Expanding both sides:
12x + 24 = 5x + 8
Subtracting 5x from both sides:
7x + 24 = 8
Subtracting 24 from both sides:
7x = -16
Dividing both sides by 7:
x = -16/7
The two equations have different solutions for x, indicating that the system has no solution.
Having no solution means that the lines represented by the equations are parallel and do not intersect.
Question 4: One Solution
First equation: 2x + 3 = 5x + 8
Subtracting 2x from both sides:
3 = 3x + 8
Subtracting 8 from both sides:
-5 = 3x
Dividing both sides by 3:
-5/3 = x
Second equation: 7x + 3 = 5x + 4
Subtracting 5x from both sides:
2x + 3 = 4
Subtracting 3 from both sides:
2x = 1
Dividing both sides by 2:
x = 1/2
The two equations have different but specific solutions for x, indicating that the system has one unique solution.
Having one solution means that the lines represented by the equations intersect at a single point.
Question 5: Writing Linear Equations
a) System with Infinite Solutions:
x + y = 5
2x + 2y = 10
This system represents two parallel lines. Any value of x and y that satisfy the first equation will also satisfy the second equation.
A real-world scenario for this system could be planning a party where each person brings an equal number of snacks (x) and drinks (y). The total number of snacks and drinks should be 10.
b) System with No Solution:
3x - 2y = 4
6x - 4y = 8
This system represents two parallel lines with the same slope. The equations are multiples of each other.
A real-world scenario for this system could be buying items at a store with two different pricing plans. The prices are inconsistent and cannot be resolved.
c) System with One Solution:
2x + 3y = 8
4x - y = 5
To solve this system, we can use either elimination or substitution method.
Using the substitution method:
From the second equation, we isolate y:
y = 4x - 5
Substituting this into the first equation:
2x + 3(4x - 5) = 8
2x + 12x - 15 = 8
14x - 15 = 8
14x = 23
x = 23/14
Substituting the value of x into the second equation:
4(23/14) - y = 5
92/14 - y = 5
92 - 14y = 5 * 14
-14y = 70 - 92
-14y = -22
y = -22/-14
So the solution to the system is x = 23/14 and y = -22/14.
A real-world scenario for this system could be solving for the number of apples (x) and oranges (y) needed to make a fruit salad. The cost of apples is twice the cost of oranges, and the budget is $8.
To investigate and explain the different types of solutions to a system of equations, we will analyze the given equations and solve them one by one.
To start with, let's look at the given set of equations:
Equation 1: 2x + 3
Equation 2: 7x + 3
Equation 3: 4x + 2
Equation 4: 5x + 5
Equation 5: 3x + 4
Equation 6: 3(4x + 8)
To determine which pairs of equations have one solution, no solution, and infinite solutions, we need to solve the equations.
Question 1:
First, let's simplify Equation 6:
Equation 6: 12x + 24
Now, we can classify the equations based on the number of solutions.
To identify the equations with infinite solutions, we look for equations that are equal after simplification. In this case, Equation 6 (12x + 24) is the same as Equation 6. So, we put a circle around Equation 6.
To identify the equations with no solution, we look for equations that contradict each other, resulting in an inconsistent system. In this case, let's compare Equation 1 (2x + 3) and Equation 2 (7x + 3). After solving them, we get:
2x + 3 = 7x + 3
-5x = 0
x = 0
Since x = 0, both equations are true. Therefore, there is no contradiction, and these equations have no solution. So, we put a square around Equation 1 and Equation 2.
To identify the equations with one solution, we look for equations that are different but still solvable with a unique solution. In this case, let's compare Equation 3 (4x + 2) and Equation 4 (5x + 5). After solving them, we get:
4x + 2 = 5x + 5
-1x = 3
x = -3
Since x = -3, the two equations have different solutions. Therefore, these equations have one solution. So, we underline Equation 3 and Equation 4.
Question 2: Infinite Solutions
Now, let's solve the equations that have infinite solutions: Equation 6.
Equation 6: 12x + 24 = 12x + 24
When we simplify Equation 6, we get the same equation on both sides. This means that any value of x will make the equation true. Therefore, the equations have infinite solutions.
To demonstrate this, let's plug in multiple values of x into the original equation:
For x = 0: 3(4(0) + 8) = 3(8) = 24
For x = 1: 3(4(1) + 8) = 3(12) = 36
For x = -5: 3(4(-5) + 8) = 3(-12) = -36
As we can see, regardless of the value of x, the equation remains true, showing that there are infinite solutions.
Question 3: No Solutions
Next, let's solve the equations that have no solution: Equation 1 and Equation 2.
Equation 1: 2x + 3 = 7x + 3
When we solve this equation, we get:
-5x = 0
x = 0
Now, let's check this solution for Equation 2: 7(0) + 3 = 3
After solving Equation 2, we get:
3 = 3
Since both equations are true for x = 0, this indicates that they have no contradiction. Therefore, there is no solution.
Question 4: One Solution
Finally, let's solve the equations that have one solution: Equation 3 and Equation 4.
Equation 3: 4x + 2 = 5x + 5
When we solve this equation, we get:
-1x = 3
x = -3
Now, let's check this solution for Equation 4: 5(-3) + 5 = 0
After solving Equation 4, we get:
0 = 0
Since both equations are true for x = -3, this indicates that they have a unique solution.
Question 5: Writing Linear Equations
To write your own systems of linear equations, consider the following examples:
1. System with Infinite Solutions:
Equation 1: 2x + 3y = 4
Equation 2: 6x + 9y = 12
Both Equations 1 and 2 represent the same line, indicating that the system has infinite solutions since the lines are parallel.
2. System with No Solution:
Equation 1: 3x + 2y = 5
Equation 2: 6x + 4y = 8
Both Equations 1 and 2 represent different lines that do not intersect, indicating that the system has no solution as there is no common point.
3. System with One Solution:
Equation 1: 2x + y = 5
Equation 2: 3x - 2y = 4
Both Equations 1 and 2 represent intersecting lines, indicating that the system has a unique solution where the lines intersect.
Remember to show your work by solving these equations to verify the solutions.
Once you have completed the worksheet with the solutions and explanations for each question, save it and submit it for a portfolio grade.