Examine the six different equations that follow determine which equations have one solution no solution or infinitely many solutions.

Question

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
2×+3=2×+7
7×+3=7×+3
4×+22=8×+10
5×+5=5×+4
3×+18=5×+8
3(4×+8)=(3×+6)
What do you notice about equations that are in each of these categories he is complete sentences to answer the questions that follow.
Consider the two equations you circled which form a system with infinite solutions solve the equation.
What do you notice about these equations?
What does it mean to have infinite solutions?
Plug in ×=3 and ×=-2 for both of the original equations to show that the equations can have more than one solution. Solve the equations. Explain your work.
Consider the two equations you put a square around which form a system with no solution solve the equations.
What does it mean to have no Solutions?
Plug in ×=2 and ×=-3 for both of the original equations to show that the equations will have no solutions. Explain your work.
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?
Use your answers to your original equations plug them into their own equations to show their Solutions are true. Use ×=4 into each original equation to prove they are false solutions. Explain your work
Now I'll try riding 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

Answer 1

Equations with infinite solutions are equations that are equivalent to each other. This means that any value of x that satisfies one equation will also satisfy the other equation. This leads to an infinite number of solutions because there are infinite values of x that can satisfy the equations.

In the system of equations 7x + 3 = 7x + 3 and 8x + 7 = 8x + 7, we notice that the equations are the same. This means that they are equivalent and have infinite solutions.

To show that these equations have infinite solutions, we can plug in x = 3 and x = -2 into both equations:

For x = 3:
7(3) + 3 = 7(3) + 3
24 = 24

For x = -2:
7(-2) + 3 = 7(-2) + 3
-13 = -13

These examples show that both equations hold true for multiple values of x.

Equations with no solutions are equations that are contradictory and cannot have any value of x that satisfies both equations.

In the system of equations 4x + 22 = 8x + 10 and 5x + 5 = 5x + 4, we notice that the equations are contradictory. This means that they do not have any values of x that can satisfy both equations.

To show that these equations have no solutions, we can plug in x = 2 and x = -3 into both equations:

For x = 2:
4(2) + 22 = 8(2) + 10
30 ≠ 26

For x = -3:
4(-3) + 22 = 8(-3) + 10
10 ≠ -14

Both examples show that the equations do not hold true for any value of x.

Equations with one solution are equations that intersect at a single point, meaning that there is only one value of x that satisfies both equations.

In the system of equations 2x + 3 = 2x + 7 and 3(4x + 8) = 3x + 6, we notice that the equations intersect at one point. This means that there is only one value of x that satisfies both equations.

To show that these equations have one solution, we can solve for x in both equations:

2x + 3 = 2x + 7
3(4x + 8) = 3x + 6

x = -2

This shows that x = -2 is the only value that satisfies both equations.

Real-world scenario: A baking company sells two types of cookies - chocolate chip and oatmeal raisin. The company sells each cookie for $1.50 and has a budget to spend $100 to purchase ingredients for the cookies. Each chocolate chip cookie requires $0.50 worth of ingredients, while each oatmeal raisin cookie requires $0.25 worth of ingredients. Write a system of equations to represent this scenario and determine the number of each type of cookie the company can make while staying within budget.

Let x represent the number of chocolate chip cookies and y represent the number of oatmeal raisin cookies.

Equation 1: 0.5x + 0.25y = 100 (total budget spent on ingredients)
Equation 2: 1.5x + 1.5y = total revenue (total revenue from selling cookies)

To find the solutions, solve the system of equations.