Question
Use the drop-down menus to complete the equations. Complete the equation so that there will be exactly one real solution. 4(3x + 2) - x = _x - 3 Complete the equation so that there will be no real solution. 4(3x + 2) - x = _x + 1 Complete the equation so that that there will be infinite real solutions. 4(3x + 2) - x = _x +_
Answers
GPT-4o mini
Let's analyze each case step by step.
1. **Exactly One Real Solution**:
To have an equation with exactly one real solution, the left side should simplify to a linear equation that equals a constant value.
Starting with:
\[
4(3x + 2) - x = _x - 3
\]
We simplify the left side:
\[
4(3x + 2) - x = 12x + 8 - x = 11x + 8
\]
Now, we want:
\[
11x + 8 = _x - 3
\]
To have exactly one solution, the right side should be a linear expression in \(x\) that is not equal to the left side. A natural choice is to choose \(11x\) so that we can set:
\[
11x + 8 = 11x - 3
\]
This would yield:
\[
8 = -3
\]
which is a contradiction. Therefore, the equation \(4(3x + 2) - x = 11x - 3\) has no solutions.
However, to ensure **one real solution**, we can set:
\[
11x + 8 = 11x + C \text{ (where } C \text{ is a constant not equal to 8)}
\]
A solid choice could be \(3x - 3\):
\[
11x + 8 = 3x - 3 \rightarrow 8 + 3 = 3x - 11x \rightarrow 11x + 8 = 3x - 3
\]
We can choose \(6\)(Any number other than 8 will ensure 1 solution) for simplicity.
Thus,:
**Answer for this part: \(3x - 3\)**
2. **No Real Solution**:
To achieve no real solutions, we want the two sides of the equation to represent parallel lines (same slope, different intercepts).
Continuing with:
\[
4(3x + 2) - x = _x + 1
\]
Similar to the previous step, we have:
\[
11x + 8 = _x + 1
\]
We can keep the left side’s slope \(11\) by maintaining an equivalent coefficient of \(x\) in the right-hand side and adjust the constant.
Therefore:
We can set:
\[
11x + 8 = 11x + 1
\]
yielding:
\[
8 = 1 \text{ , which is a contradiction.}
\] Therefore, it yields no real solutions.
Hence:
**Answer for this part: \(11x + 1\)**
3. **Infinite Real Solutions**:
For infinite solutions, both sides of the equation must be equivalent.
For:
\[
4(3x + 2) - x = _x + _
\]
As before:
\[
11x + 8 = mx + b \text{ (equate both)}
\]
So we take:
Set:
\[
11x + 8 = 11x + 8 \text{ (coefficients and constant must match)}}
\]
This can be fulfilled by matching both sides as follows:
Therefore, for infinite solutions:
**Answer for this part: \(11x + 8\)**
### Final answers:
1. Exactly one real solution: **\(3x - 3\)**
2. No real solution: **\(11x + 1\)**
3. Infinite real solutions: **\(11x + 8\)**
1. **Exactly One Real Solution**:
To have an equation with exactly one real solution, the left side should simplify to a linear equation that equals a constant value.
Starting with:
\[
4(3x + 2) - x = _x - 3
\]
We simplify the left side:
\[
4(3x + 2) - x = 12x + 8 - x = 11x + 8
\]
Now, we want:
\[
11x + 8 = _x - 3
\]
To have exactly one solution, the right side should be a linear expression in \(x\) that is not equal to the left side. A natural choice is to choose \(11x\) so that we can set:
\[
11x + 8 = 11x - 3
\]
This would yield:
\[
8 = -3
\]
which is a contradiction. Therefore, the equation \(4(3x + 2) - x = 11x - 3\) has no solutions.
However, to ensure **one real solution**, we can set:
\[
11x + 8 = 11x + C \text{ (where } C \text{ is a constant not equal to 8)}
\]
A solid choice could be \(3x - 3\):
\[
11x + 8 = 3x - 3 \rightarrow 8 + 3 = 3x - 11x \rightarrow 11x + 8 = 3x - 3
\]
We can choose \(6\)(Any number other than 8 will ensure 1 solution) for simplicity.
Thus,:
**Answer for this part: \(3x - 3\)**
2. **No Real Solution**:
To achieve no real solutions, we want the two sides of the equation to represent parallel lines (same slope, different intercepts).
Continuing with:
\[
4(3x + 2) - x = _x + 1
\]
Similar to the previous step, we have:
\[
11x + 8 = _x + 1
\]
We can keep the left side’s slope \(11\) by maintaining an equivalent coefficient of \(x\) in the right-hand side and adjust the constant.
Therefore:
We can set:
\[
11x + 8 = 11x + 1
\]
yielding:
\[
8 = 1 \text{ , which is a contradiction.}
\] Therefore, it yields no real solutions.
Hence:
**Answer for this part: \(11x + 1\)**
3. **Infinite Real Solutions**:
For infinite solutions, both sides of the equation must be equivalent.
For:
\[
4(3x + 2) - x = _x + _
\]
As before:
\[
11x + 8 = mx + b \text{ (equate both)}
\]
So we take:
Set:
\[
11x + 8 = 11x + 8 \text{ (coefficients and constant must match)}}
\]
This can be fulfilled by matching both sides as follows:
Therefore, for infinite solutions:
**Answer for this part: \(11x + 8\)**
### Final answers:
1. Exactly one real solution: **\(3x - 3\)**
2. No real solution: **\(11x + 1\)**
3. Infinite real solutions: **\(11x + 8\)**