2x+3=2x+7 (no solution)
7x+3=7x+3 (infinite solutions)
4x+22=8x+10 (no solution)
5x+5=5x+4 (no solution)
3x+18=5x+8 (one solution)
3(4x+8)=4(3x+6) (infinite solutions)
Examine the six different equations that follow. Determine which pairs of equations
have one solution, no solution, and infinite solutions.
• 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
7x+3=7x+3
4x+22=8x+10
5x+5=5x+4
3x+18=5x+8
3(4x+8)=4(3x+6)
11 answers
What do you notice about equations that are in each of these categories? Use
complete sentences to answer the questions that follow.
complete sentences to answer the questions that follow.
Equations that have no solution typically involve situations where contradictory statements are being compared, such as 2x+3=2x+7. These equations do not have a common value for the variable that would satisfy both sides of the equation.
Equations that have infinite solutions usually involve situations where the expressions on both sides of the equation are equivalent, such as 7x+3=7x+3. In these cases, any value of the variable can be substituted into the equation and it will hold true.
Equations that have one solution involve scenarios where the expressions on each side of the equation can be simplified to a single common value, such as 3x+18=5x+8. This results in a unique solution for the variable that satisfies the equation.
Equations that have infinite solutions usually involve situations where the expressions on both sides of the equation are equivalent, such as 7x+3=7x+3. In these cases, any value of the variable can be substituted into the equation and it will hold true.
Equations that have one solution involve scenarios where the expressions on each side of the equation can be simplified to a single common value, such as 3x+18=5x+8. This results in a unique solution for the variable that satisfies the equation.
put that in one answer
Equations with no solution involve comparing contradictory statements, such as 2x+3=2x+7, leading to no common value for the variable. Equations with infinite solutions feature equivalent expressions on both sides, like 7x+3=7x+3, where any value for the variable satisfies the equation. Equations with one solution, such as 3x+18=5x+8, have expressions that simplify to a unique value for the variable that satisfies the equation.
Consider the two equations you circled, which form a system with infinite solutions.
Solve the equations.
Solve the equations.
The two equations that form a system with infinite solutions are:
1. 7x + 3 = 7x + 3 (equation 1)
2. 3(4x + 8) = 4(3x + 6) (equation 2)
For Equation 1:
7x + 3 = 7x + 3
This equation simplifies to 0 = 0. Since this equation is an identity, it indicates that the two sides are equal regardless of the value of x. This means there are an infinite number of solutions to this equation.
For Equation 2:
3(4x + 8) = 4(3x + 6)
12x + 24 = 12x + 24
Subtracting 12x from both sides:
24 = 24
Again, this equation simplifies to 24 = 24. As with equation 1, this equation is an identity, showing that it also has infinite solutions.
Therefore, the system of equations formed by these two equations has an infinity of solutions and is consistent.
1. 7x + 3 = 7x + 3 (equation 1)
2. 3(4x + 8) = 4(3x + 6) (equation 2)
For Equation 1:
7x + 3 = 7x + 3
This equation simplifies to 0 = 0. Since this equation is an identity, it indicates that the two sides are equal regardless of the value of x. This means there are an infinite number of solutions to this equation.
For Equation 2:
3(4x + 8) = 4(3x + 6)
12x + 24 = 12x + 24
Subtracting 12x from both sides:
24 = 24
Again, this equation simplifies to 24 = 24. As with equation 1, this equation is an identity, showing that it also has infinite solutions.
Therefore, the system of equations formed by these two equations has an infinity of solutions and is consistent.
what do you notice about these equations?
what does it mean to have infinite solutions?
what does it mean to have infinite solutions?
In the two equations given, 7x + 3 = 7x + 3 and 3(4x + 8) = 4(3x + 6), both reduce to identities rather than distinct equations with a single, finite solution. An identity means that the equation is true for any value of the variable x, it does not restrict the variable to certain values as it remains true regardless.
Having infinite solutions means that the equations provide a range of valid solutions, often covering an infinite set of values that satisfy the given equations. This condition occurs when the equations are essentially the same or can be reduced to the same expression through algebraic manipulation. It indicates that there is not just one particular solution to the system of equations but many potential solutions that satisfy the conditions set by the equations.
Having infinite solutions means that the equations provide a range of valid solutions, often covering an infinite set of values that satisfy the given equations. This condition occurs when the equations are essentially the same or can be reduced to the same expression through algebraic manipulation. It indicates that there is not just one particular solution to the system of equations but many potential solutions that satisfy the conditions set by the equations.
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
equations can have more than one solution. Explain your work
Let's plug in two different points to the equations that have infinite solutions.
1. 7x + 3 = 7x + 3
Let's solve this equation with x = 0:
7(0) + 3 = 7(0) + 3
3 = 3, the equation holds true for x = 0.
Now, let's solve it with x = 5:
7(5) + 3 = 7(5) + 3
35 + 3 = 35 + 3
38 = 38, the equation also holds true for x = 5.
2. 3(4x + 8) = 4(3x + 6)
Let's solve this equation with x = 1:
3(4(1) + 8) = 4(3(1) + 6)
3(4 + 8) = 4(3 + 6)
3(12) = 4(9)
36 = 36, the equation holds true for x = 1.
Next, we'll solve it with x = –2:
3(4(-2) + 8) = 4(3(-2) + 6)
3(-8 + 8) = 4(-6 + 6)
3(0) = 4(0)
0 = 0, the equation also holds true for x = –2.
By plugging in different values for x, such as 0, 5, 1, and −2 in both equations, we see that each equation holds true for various values of x. This demonstrates that the equations have more than one solution, showing the existence of an infinite number of solutions to the system.
1. 7x + 3 = 7x + 3
Let's solve this equation with x = 0:
7(0) + 3 = 7(0) + 3
3 = 3, the equation holds true for x = 0.
Now, let's solve it with x = 5:
7(5) + 3 = 7(5) + 3
35 + 3 = 35 + 3
38 = 38, the equation also holds true for x = 5.
2. 3(4x + 8) = 4(3x + 6)
Let's solve this equation with x = 1:
3(4(1) + 8) = 4(3(1) + 6)
3(4 + 8) = 4(3 + 6)
3(12) = 4(9)
36 = 36, the equation holds true for x = 1.
Next, we'll solve it with x = –2:
3(4(-2) + 8) = 4(3(-2) + 6)
3(-8 + 8) = 4(-6 + 6)
3(0) = 4(0)
0 = 0, the equation also holds true for x = –2.
By plugging in different values for x, such as 0, 5, 1, and −2 in both equations, we see that each equation holds true for various values of x. This demonstrates that the equations have more than one solution, showing the existence of an infinite number of solutions to the system.
17 answers