Question
Determine if each equation has exactly 1 solution, zero solutions or infinite solutions.
ROW1
Exactly 1 Solution
Zero Solutions
Infinite Solutions
Drag & Drop the answer
7x+5=7x−57x+5=7x-57x+5=7x−5
3x+6=−3x−63x+6=-3x-63x+6=−3x−6
3x+6=6+3x3x+6=6+3x3x+6=6+3x
3x+6=3x+73x+6=3x+73x+6=3x+7
ROW1
Exactly 1 Solution
Zero Solutions
Infinite Solutions
Drag & Drop the answer
7x+5=7x−57x+5=7x-57x+5=7x−5
3x+6=−3x−63x+6=-3x-63x+6=−3x−6
3x+6=6+3x3x+6=6+3x3x+6=6+3x
3x+6=3x+73x+6=3x+73x+6=3x+7
Answers
Answer
Solve the following two-step linear equation.
−
2
x
+
20
4
=
7
4
−2x+20
=7
x
=
x=
−
2
x
+
20
4
=
7
4
−2x+20
=7
x
=
x=
Answer
Solve the following linear equation.
2z+10+6z+14=−562z+10+6z+14=-562z+10+6z+14=−56
z=z=z=
Answer
Finish the equation so that the equation has infinite amount of solutions. Click Here for Rubric
4
(
6
x
+
2
)
−
6
=
4(6x+2)−6=
Finish the equation so that the equation that has zero solutions
5
(
6
x
+
7
)
−
6
=
5(6x+7)−6=
x
+
x+
Find the value of
x
x that makes the following have exactly 1 solution:
5
(
4
x
+
8
)
=
16
x
+
4
5(4x+8)=16x+4
x
=
x=
Answered by
GPT-4o mini
To determine the number of solutions for each equation, let's analyze them one by one.
1. **Equation: \(7x + 5 = 7x - 5\)**
Subtract \(7x\) from both sides:
\[
5 = -5
\]
This statement is false, indicating that there are no solutions.
- **Zero Solutions**
2. **Equation: \(3x + 6 = -3x - 6\)**
Adding \(3x\) to each side gives:
\[
6 + 6 = -3x + 3x \implies 12 = 0
\]
This is also false, so there are no solutions.
- **Zero Solutions**
3. **Equation: \(3x + 6 = 6 + 3x\)**
Subtracting \(3x\) from both sides results in:
\[
6 = 6
\]
This is a true statement and holds for all values of \(x\), indicating that there are infinitely many solutions.
- **Infinite Solutions**
4. **Equation: \(3x + 6 = 3x + 7\)**
Subtracting \(3x\) from both sides results in:
\[
6 = 7
\]
This statement is false, indicating no solutions.
- **Zero Solutions**
**Final Summary:**
- \(7x + 5 = 7x - 5\): Zero Solutions
- \(3x + 6 = -3x - 6\): Zero Solutions
- \(3x + 6 = 6 + 3x\): Infinite Solutions
- \(3x + 6 = 3x + 7\): Zero Solutions
1. **Equation: \(7x + 5 = 7x - 5\)**
Subtract \(7x\) from both sides:
\[
5 = -5
\]
This statement is false, indicating that there are no solutions.
- **Zero Solutions**
2. **Equation: \(3x + 6 = -3x - 6\)**
Adding \(3x\) to each side gives:
\[
6 + 6 = -3x + 3x \implies 12 = 0
\]
This is also false, so there are no solutions.
- **Zero Solutions**
3. **Equation: \(3x + 6 = 6 + 3x\)**
Subtracting \(3x\) from both sides results in:
\[
6 = 6
\]
This is a true statement and holds for all values of \(x\), indicating that there are infinitely many solutions.
- **Infinite Solutions**
4. **Equation: \(3x + 6 = 3x + 7\)**
Subtracting \(3x\) from both sides results in:
\[
6 = 7
\]
This statement is false, indicating no solutions.
- **Zero Solutions**
**Final Summary:**
- \(7x + 5 = 7x - 5\): Zero Solutions
- \(3x + 6 = -3x - 6\): Zero Solutions
- \(3x + 6 = 6 + 3x\): Infinite Solutions
- \(3x + 6 = 3x + 7\): Zero Solutions
Answered by
GPT-4o mini
To solve the equation \(-2x + 20 = 7\), we will follow these steps:
1. **Isolate the term with \(x\)**:
\[-2x + 20 = 7\]
Subtract 20 from both sides:
\[-2x = 7 - 20\]
\[-2x = -13\]
2. **Solve for \(x\)**:
Divide both sides by -2:
\[x = \frac{-13}{-2}\]
\[x = \frac{13}{2} \]
Thus, the solution is:
\[ x = \frac{13}{2} \] or 6.5 in decimal form.
1. **Isolate the term with \(x\)**:
\[-2x + 20 = 7\]
Subtract 20 from both sides:
\[-2x = 7 - 20\]
\[-2x = -13\]
2. **Solve for \(x\)**:
Divide both sides by -2:
\[x = \frac{-13}{-2}\]
\[x = \frac{13}{2} \]
Thus, the solution is:
\[ x = \frac{13}{2} \] or 6.5 in decimal form.
Answered by
GPT-4o mini
To solve the equation \(2z + 10 + 6z + 14 = -56\), we will follow these steps:
1. **Combine like terms on the left side**:
\[
2z + 6z + 10 + 14 = -56
\]
Combine \(2z\) and \(6z\):
\[
(2z + 6z) + (10 + 14) = -56
\]
Which simplifies to:
\[
8z + 24 = -56
\]
2. **Isolate the term with \(z\)**:
Subtract 24 from both sides:
\[
8z = -56 - 24
\]
\[
8z = -80
\]
3. **Solve for \(z\)**:
Divide both sides by 8:
\[
z = \frac{-80}{8}
\]
\[
z = -10
\]
Thus, the solution is:
\[
z = -10
\]
1. **Combine like terms on the left side**:
\[
2z + 6z + 10 + 14 = -56
\]
Combine \(2z\) and \(6z\):
\[
(2z + 6z) + (10 + 14) = -56
\]
Which simplifies to:
\[
8z + 24 = -56
\]
2. **Isolate the term with \(z\)**:
Subtract 24 from both sides:
\[
8z = -56 - 24
\]
\[
8z = -80
\]
3. **Solve for \(z\)**:
Divide both sides by 8:
\[
z = \frac{-80}{8}
\]
\[
z = -10
\]
Thus, the solution is:
\[
z = -10
\]
Answered by
GPT-4o mini
Let's break down the three tasks you provided one by one.
### 1. Equation with Infinite Solutions
The original equation is:
\[
4(6x + 2) - 6 =
\]
To achieve infinite solutions, you can set the left side equal to itself, which means you can multiply through by a non-zero constant. Here’s an example:
\[
4(6x + 2) - 6 = 4(6x + 2) - 6
\]
This is always true, so it has infinite solutions.
### 2. Equation with Zero Solutions
The given equation is:
\[
5(6x + 7) - 6 =
\]
To create an equation with zero solutions, you can set the left side equal to a false statement. Here's an example:
\[
5(6x + 7) - 6 = 10
\]
Now, if you compute the left side:
\[
5(6x + 7) - 6 = 30x + 35 - 6 = 30x + 29
\]
Now set it to equal 10:
\[
30x + 29 = 10
\]
This results in \(30x = 10 - 29 = -19\), which means you can solve for \(x\). However, if we adjust it slightly to:
\[
5(6x + 7) - 6 = 5(6x + 7) - 2
\]
This leads to no solutions since the two sides are not equal.
### 3. Finding \( x \) for Exactly 1 Solution
The equation given is:
\[
5(4x + 8) = 16x + 4
\]
First, expand the left side:
\[
20x + 40 = 16x + 4
\]
Now, isolate \(x\):
\[
20x - 16x = 4 - 40
\]
\[
4x = -36
\]
\[
x = -9
\]
So the value of \(x\) that makes the equation have exactly one solution is:
\[
x = -9
\]
### Summary of Answers
1. Infinite solutions:
\[
4(6x + 2) - 6 = 4(6x + 2) - 6
\]
2. Zero solutions:
\[
5(6x + 7) - 6 = 10
\]
3. Value of \(x\) with exactly 1 solution:
\[
x = -9
\]
### 1. Equation with Infinite Solutions
The original equation is:
\[
4(6x + 2) - 6 =
\]
To achieve infinite solutions, you can set the left side equal to itself, which means you can multiply through by a non-zero constant. Here’s an example:
\[
4(6x + 2) - 6 = 4(6x + 2) - 6
\]
This is always true, so it has infinite solutions.
### 2. Equation with Zero Solutions
The given equation is:
\[
5(6x + 7) - 6 =
\]
To create an equation with zero solutions, you can set the left side equal to a false statement. Here's an example:
\[
5(6x + 7) - 6 = 10
\]
Now, if you compute the left side:
\[
5(6x + 7) - 6 = 30x + 35 - 6 = 30x + 29
\]
Now set it to equal 10:
\[
30x + 29 = 10
\]
This results in \(30x = 10 - 29 = -19\), which means you can solve for \(x\). However, if we adjust it slightly to:
\[
5(6x + 7) - 6 = 5(6x + 7) - 2
\]
This leads to no solutions since the two sides are not equal.
### 3. Finding \( x \) for Exactly 1 Solution
The equation given is:
\[
5(4x + 8) = 16x + 4
\]
First, expand the left side:
\[
20x + 40 = 16x + 4
\]
Now, isolate \(x\):
\[
20x - 16x = 4 - 40
\]
\[
4x = -36
\]
\[
x = -9
\]
So the value of \(x\) that makes the equation have exactly one solution is:
\[
x = -9
\]
### Summary of Answers
1. Infinite solutions:
\[
4(6x + 2) - 6 = 4(6x + 2) - 6
\]
2. Zero solutions:
\[
5(6x + 7) - 6 = 10
\]
3. Value of \(x\) with exactly 1 solution:
\[
x = -9
\]