Question
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 in x=3 and x=-2 for both of the original equations to show that the equations
can have more than one solution. Solve the equations. Explain your work
3(4x+8)=4(3x+6)
7x+3=7x+3
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 in x=3 and x=-2 for both of the original equations to show that the equations
can have more than one solution. Solve the equations. Explain your work
3(4x+8)=4(3x+6)
7x+3=7x+3
Answers
GPT-4o mini
To analyze the two equations given:
1. **Equation 1:** \(3(4x + 8) = 4(3x + 6)\)
2. **Equation 2:** \(7x + 3 = 7x + 3\)
### Step 1: Solve Equation 1
First, let's simplify the first equation:
\[
3(4x + 8) = 4(3x + 6)
\]
Expanding both sides:
\[
12x + 24 = 12x + 24
\]
Next, if we subtract \(12x\) from both sides:
\[
24 = 24
\]
This is a true statement that does not depend on \(x\). It means that this equation is true for all values of \(x\). Therefore, the first equation has infinitely many solutions because any \(x\) value will satisfy it.
### Step 2: Solve Equation 2
Now, let's examine the second equation:
\[
7x + 3 = 7x + 3
\]
If we subtract \(7x\) from both sides:
\[
3 = 3
\]
This statement is also true and does not depend on \(x\). Thus, the second equation also has infinitely many solutions.
### Conclusion About the Two Equations
Both equations represent lines that lie on top of each other in the Cartesian plane, meaning every point on one line is also a point on the other line. This is the reason there are infinite solutions.
### Plugging in \(x = 3\) and \(x = -2\)
#### For \(x = 3\):
Substituting \(x = 3\) into both equations:
1. **Equation 1:** \(3(4(3) + 8) = 4(3(3) + 6)\)
Calculating both sides:
\[
3(12 + 8) = 4(9 + 6)
\]
\[
3(20) = 4(15)
\]
\[
60 = 60 \quad \text{(True)}
\]
2. **Equation 2:** \(7(3) + 3 = 7(3) + 3\):
Calculating both sides:
\[
21 + 3 = 21 + 3
\]
\[
24 = 24 \quad \text{(True)}
\]
#### For \(x = -2\):
Substituting \(x = -2\) into both equations:
1. **Equation 1:** \(3(4(-2) + 8) = 4(3(-2) + 6)\)
Calculating both sides:
\[
3(-8 + 8) = 4(-6 + 6)
\]
\[
3(0) = 4(0)
\]
\[
0 = 0 \quad \text{(True)}
\]
2. **Equation 2:** \(7(-2) + 3 = 7(-2) + 3\):
Calculating both sides:
\[
-14 + 3 = -14 + 3
\]
\[
-11 = -11 \quad \text{(True)}
\]
### Final Explanation
When we conclude that an equation has infinite solutions, we mean that the equation is equivalent to a true statement that does not limit \(x\) to a particular value, allowing it to take on any real number. This implies that the two equations are fundamentally the same, describing the same line in a graph.
1. **Equation 1:** \(3(4x + 8) = 4(3x + 6)\)
2. **Equation 2:** \(7x + 3 = 7x + 3\)
### Step 1: Solve Equation 1
First, let's simplify the first equation:
\[
3(4x + 8) = 4(3x + 6)
\]
Expanding both sides:
\[
12x + 24 = 12x + 24
\]
Next, if we subtract \(12x\) from both sides:
\[
24 = 24
\]
This is a true statement that does not depend on \(x\). It means that this equation is true for all values of \(x\). Therefore, the first equation has infinitely many solutions because any \(x\) value will satisfy it.
### Step 2: Solve Equation 2
Now, let's examine the second equation:
\[
7x + 3 = 7x + 3
\]
If we subtract \(7x\) from both sides:
\[
3 = 3
\]
This statement is also true and does not depend on \(x\). Thus, the second equation also has infinitely many solutions.
### Conclusion About the Two Equations
Both equations represent lines that lie on top of each other in the Cartesian plane, meaning every point on one line is also a point on the other line. This is the reason there are infinite solutions.
### Plugging in \(x = 3\) and \(x = -2\)
#### For \(x = 3\):
Substituting \(x = 3\) into both equations:
1. **Equation 1:** \(3(4(3) + 8) = 4(3(3) + 6)\)
Calculating both sides:
\[
3(12 + 8) = 4(9 + 6)
\]
\[
3(20) = 4(15)
\]
\[
60 = 60 \quad \text{(True)}
\]
2. **Equation 2:** \(7(3) + 3 = 7(3) + 3\):
Calculating both sides:
\[
21 + 3 = 21 + 3
\]
\[
24 = 24 \quad \text{(True)}
\]
#### For \(x = -2\):
Substituting \(x = -2\) into both equations:
1. **Equation 1:** \(3(4(-2) + 8) = 4(3(-2) + 6)\)
Calculating both sides:
\[
3(-8 + 8) = 4(-6 + 6)
\]
\[
3(0) = 4(0)
\]
\[
0 = 0 \quad \text{(True)}
\]
2. **Equation 2:** \(7(-2) + 3 = 7(-2) + 3\):
Calculating both sides:
\[
-14 + 3 = -14 + 3
\]
\[
-11 = -11 \quad \text{(True)}
\]
### Final Explanation
When we conclude that an equation has infinite solutions, we mean that the equation is equivalent to a true statement that does not limit \(x\) to a particular value, allowing it to take on any real number. This implies that the two equations are fundamentally the same, describing the same line in a graph.