Question
Solve and Determine if the equation has one, none or infinite solutions.
-11 + x = -7x - 8(-x + 1)
33 + 6x = 3(-1 + 5x)
-5(x - 1) = 5 - 5x
12 + 4n = 4(n + 3)
4(-4 - 8m) + 28m + 4m = -272
-11 + x = -7x - 8(-x + 1)
33 + 6x = 3(-1 + 5x)
-5(x - 1) = 5 - 5x
12 + 4n = 4(n + 3)
4(-4 - 8m) + 28m + 4m = -272
Answers
GPT-4o mini
Let's solve each equation step by step and determine the number of solutions for each.
**1. Equation: \(-11 + x = -7x - 8(-x + 1)\)**
First, simplify the right side:
\[
-11 + x = -7x + 8x - 8
\]
\[
-11 + x = x - 8
\]
Now, subtract \(x\) from both sides:
\[
-11 = -8
\]
Since \(-11\) does not equal \(-8\), this equation has **no solutions**.
---
**2. Equation: \(33 + 6x = 3(-1 + 5x)\)**
Simplify the right side:
\[
33 + 6x = -3 + 15x
\]
Rearranging gives:
\[
33 + 3 = 15x - 6x
\]
\[
36 = 9x
\]
\[
x = 4
\]
This equation has **one solution**: \(x = 4\).
---
**3. Equation: \(-5(x - 1) = 5 - 5x\)**
Expanding the left side:
\[
-5x + 5 = 5 - 5x
\]
Adding \(5x\) to both sides:
\[
5 = 5
\]
Since this is always true, this equation has **infinite solutions**.
---
**4. Equation: \(12 + 4n = 4(n + 3)\)**
Expanding the right side:
\[
12 + 4n = 4n + 12
\]
Subtracting \(4n\) from both sides:
\[
12 = 12
\]
Again, this is always true, so this equation has **infinite solutions**.
---
**5. Equation: \(4(-4 - 8m) + 28m + 4m = -272\)**
First, distribute on the left side:
\[
-16 - 32m + 28m + 4m = -272
\]
Combining like terms:
\[
-16 = -272
\]
Since \(-16\) does not equal \(-272\), this equation has **no solutions**.
---
### Summary of Solutions:
1. **No solutions**
2. **One solution**: \(x = 4\)
3. **Infinite solutions**
4. **Infinite solutions**
5. **No solutions**
**1. Equation: \(-11 + x = -7x - 8(-x + 1)\)**
First, simplify the right side:
\[
-11 + x = -7x + 8x - 8
\]
\[
-11 + x = x - 8
\]
Now, subtract \(x\) from both sides:
\[
-11 = -8
\]
Since \(-11\) does not equal \(-8\), this equation has **no solutions**.
---
**2. Equation: \(33 + 6x = 3(-1 + 5x)\)**
Simplify the right side:
\[
33 + 6x = -3 + 15x
\]
Rearranging gives:
\[
33 + 3 = 15x - 6x
\]
\[
36 = 9x
\]
\[
x = 4
\]
This equation has **one solution**: \(x = 4\).
---
**3. Equation: \(-5(x - 1) = 5 - 5x\)**
Expanding the left side:
\[
-5x + 5 = 5 - 5x
\]
Adding \(5x\) to both sides:
\[
5 = 5
\]
Since this is always true, this equation has **infinite solutions**.
---
**4. Equation: \(12 + 4n = 4(n + 3)\)**
Expanding the right side:
\[
12 + 4n = 4n + 12
\]
Subtracting \(4n\) from both sides:
\[
12 = 12
\]
Again, this is always true, so this equation has **infinite solutions**.
---
**5. Equation: \(4(-4 - 8m) + 28m + 4m = -272\)**
First, distribute on the left side:
\[
-16 - 32m + 28m + 4m = -272
\]
Combining like terms:
\[
-16 = -272
\]
Since \(-16\) does not equal \(-272\), this equation has **no solutions**.
---
### Summary of Solutions:
1. **No solutions**
2. **One solution**: \(x = 4\)
3. **Infinite solutions**
4. **Infinite solutions**
5. **No solutions**
Define each of these
One solution, No Solutions, Infinite solutions
One solution, No Solutions, Infinite solutions
GPT-4o mini
Certainly! Here are the definitions for each of the terms:
### 1. One Solution
**Definition**: An equation has one solution when there is exactly one value of the variable that makes the equation true. In the context of linear equations, this typically means that the lines represented by the two sides of the equation intersect at exactly one point. Graphically, this is represented as two distinct lines crossing each other at a single point.
**Example**: The equation \(x + 3 = 7\) has one solution, \(x = 4\).
---
### 2. No Solutions
**Definition**: An equation has no solutions when there is no value of the variable that can satisfy the equation. This occurs when the two sides of the equation represent parallel lines that never intersect. As a result, there are no points at which the equation is true.
**Example**: The equation \(2x + 1 = 2x + 5\) has no solutions because simplifying leads to \(1 = 5\), which is a contradiction.
---
### 3. Infinite Solutions
**Definition**: An equation has infinite solutions when there are multiple (in fact, an unlimited number of) values of the variable that satisfy the equation. This generally occurs when one side of the equation can be transformed to become identical to the other side, meaning the two representations are equivalent for all values of the variable. Graphically, this is represented as the same line on both sides of the equation.
**Example**: The equation \(3(x - 2) = 3x - 6\) has infinite solutions because both sides simplify to the same expression for all values of \(x\).
---
These definitions encapsulate the different possibilities encountered when solving equations in algebra.
### 1. One Solution
**Definition**: An equation has one solution when there is exactly one value of the variable that makes the equation true. In the context of linear equations, this typically means that the lines represented by the two sides of the equation intersect at exactly one point. Graphically, this is represented as two distinct lines crossing each other at a single point.
**Example**: The equation \(x + 3 = 7\) has one solution, \(x = 4\).
---
### 2. No Solutions
**Definition**: An equation has no solutions when there is no value of the variable that can satisfy the equation. This occurs when the two sides of the equation represent parallel lines that never intersect. As a result, there are no points at which the equation is true.
**Example**: The equation \(2x + 1 = 2x + 5\) has no solutions because simplifying leads to \(1 = 5\), which is a contradiction.
---
### 3. Infinite Solutions
**Definition**: An equation has infinite solutions when there are multiple (in fact, an unlimited number of) values of the variable that satisfy the equation. This generally occurs when one side of the equation can be transformed to become identical to the other side, meaning the two representations are equivalent for all values of the variable. Graphically, this is represented as the same line on both sides of the equation.
**Example**: The equation \(3(x - 2) = 3x - 6\) has infinite solutions because both sides simplify to the same expression for all values of \(x\).
---
These definitions encapsulate the different possibilities encountered when solving equations in algebra.