Math / Algebra

QuestionWhich of the following equations hav exactly one solution?
Choose all answers that apply: A $2 x-31=2 x-31$ B $2 x-31=-2 x-31$ (c) $2 x+31=2 x-31$ (D) $2 x-2=2 x-31$

Studdy Solution

STEP 1

What is this asking? Which of these equations have only one solution for $x$ ? Watch out! Some equations can have no solutions or infinitely many solutions!
Don't fall for that trap!

STEP 2

1. Analyze Equation A
2. Analyze Equation B
3. Analyze Equation C
4. Analyze Equation D

STEP 3

Alright, let's start with equation A: $2x - 31 = 2x - 31$ .
This looks interesting!

STEP 4

Notice that both sides of the equation are exactly the same.
This means no matter what value we plug in for $x$ , the equation will always be true!

STEP 5

For example, if $x = 0$ , we get $-31 = -31$ .
If $x = 1$ , we get $-29 = -29$ .
See? Always true!
So, equation A has infinitely many solutions, not just one.

STEP 6

Now, let's move on to equation B: $2x - 31 = -2x - 31$ .

STEP 7

Let's add $2x$ to both sides to group the $x$ terms: $2x - 31 + 2x = -2x - 31 + 2x$ , which simplifies to $4x - 31 = -31$ .

STEP 8

Next, let's add 31 to both sides to isolate the $x$ term: $4x - 31 + 31 = -31 + 31$ , which simplifies to $4x = 0$ .

STEP 9

Finally, let's divide both sides by 4 to solve for $x$ : $4x \cdot \frac{1}{4} = 0 \cdot \frac{1}{4}$ , which gives us $x = 0$ .
One solution!
Awesome!

STEP 10

Time for equation C: $2x + 31 = 2x - 31$ .

STEP 11

Let's subtract $2x$ from both sides: $2x + 31 - 2x = 2x - 31 - 2x$ , which simplifies to $31 = -31$ .
Uh oh!
That's not right!

STEP 12

Since 31 is never equal to -31, this equation has no solutions.
Tricky!

STEP 13

Last but not least, equation D: $2x - 2 = 2x - 31$ .

STEP 14

Let's subtract $2x$ from both sides: $2x - 2 - 2x = 2x - 31 - 2x$ , which simplifies to $-2 = -31$ .
Nope, that's not true either!

STEP 15

So, equation D also has no solutions.

STEP 16

Only equation B, $2x - 31 = -2x - 31$ , has exactly one solution, which is $x = 0$ .

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QuestionWhich of the following equations hav exactly one solution?
Choose all answers that apply: A $2 x-31=2 x-31$ B $2 x-31=-2 x-31$ (c) $2 x+31=2 x-31$ (D) $2 x-2=2 x-31$

Studdy Solution

STEP 1

What is this asking? Which of these equations have only one solution for $x$ ? Watch out! Some equations can have no solutions or infinitely many solutions!
Don't fall for that trap!

STEP 2

1. Analyze Equation A
2. Analyze Equation B
3. Analyze Equation C
4. Analyze Equation D

STEP 3

Alright, let's start with equation A: $2x - 31 = 2x - 31$ .
This looks interesting!

STEP 4

Notice that both sides of the equation are exactly the same.
This means no matter what value we plug in for $x$ , the equation will always be true!

STEP 5

For example, if $x = 0$ , we get $-31 = -31$ .
If $x = 1$ , we get $-29 = -29$ .
See? Always true!
So, equation A has infinitely many solutions, not just one.

STEP 6

Now, let's move on to equation B: $2x - 31 = -2x - 31$ .

STEP 7

Let's add $2x$ to both sides to group the $x$ terms: $2x - 31 + 2x = -2x - 31 + 2x$ , which simplifies to $4x - 31 = -31$ .

STEP 8

Next, let's add 31 to both sides to isolate the $x$ term: $4x - 31 + 31 = -31 + 31$ , which simplifies to $4x = 0$ .

STEP 9

Finally, let's divide both sides by 4 to solve for $x$ : $4x \cdot \frac{1}{4} = 0 \cdot \frac{1}{4}$ , which gives us $x = 0$ .
One solution!
Awesome!

STEP 10

Time for equation C: $2x + 31 = 2x - 31$ .

STEP 11

Let's subtract $2x$ from both sides: $2x + 31 - 2x = 2x - 31 - 2x$ , which simplifies to $31 = -31$ .
Uh oh!
That's not right!

STEP 12

Since 31 is never equal to -31, this equation has no solutions.
Tricky!

STEP 13

Last but not least, equation D: $2x - 2 = 2x - 31$ .

STEP 14

Let's subtract $2x$ from both sides: $2x - 2 - 2x = 2x - 31 - 2x$ , which simplifies to $-2 = -31$ .
Nope, that's not true either!

STEP 15

So, equation D also has no solutions.

STEP 16

Only equation B, $2x - 31 = -2x - 31$ , has exactly one solution, which is $x = 0$ .

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QuestionWhich of the following equations hav exactly one solution?Choose all answers that apply: A 2x−31=2x−312 x-31=2 x-312x−31=2x−31 B 2x−31=−2x−312 x-31=-2 x-312x−31=−2x−31 (c) 2x+31=2x−312 x+31=2 x-312x+31=2x−31 (D) 2x−2=2x−312 x-2=2 x-312x−2=2x−31

Studdy Solution

STEP 1

What is this asking? Which of these equations have only *one* solution for xxx? Watch out! Some equations can have *no* solutions or *infinitely many* solutions!Don't fall for that trap!

STEP 2

1. Analyze Equation A2. Analyze Equation B3. Analyze Equation C4. Analyze Equation D

STEP 3

Alright, let's **start** with equation A: 2x−31=2x−312x - 31 = 2x - 312x−31=2x−31.This looks interesting!

STEP 4

Notice that both sides of the equation are *exactly* the same.This means no matter what value we plug in for xxx, the equation will *always* be true!

STEP 5

For example, if x=0x = 0x=0, we get −31=−31-31 = -31−31=−31.If x=1x = 1x=1, we get −29=−29-29 = -29−29=−29.See? Always true!So, equation A has **infinitely many** solutions, not just one.

STEP 6

Now, let's **move on** to equation B: 2x−31=−2x−312x - 31 = -2x - 312x−31=−2x−31.

STEP 7

Let's **add** 2x2x2x to both sides to **group** the xxx terms: 2x−31+2x=−2x−31+2x2x - 31 + 2x = -2x - 31 + 2x2x−31+2x=−2x−31+2x, which simplifies to 4x−31=−314x - 31 = -314x−31=−31.

STEP 8

Next, let's **add** 31 to both sides to **isolate** the xxx term: 4x−31+31=−31+314x - 31 + 31 = -31 + 314x−31+31=−31+31, which simplifies to 4x=04x = 04x=0.

STEP 9

Finally, let's **divide** both sides by 4 to **solve** for xxx: 4x⋅14=0⋅144x \cdot \frac{1}{4} = 0 \cdot \frac{1}{4}4x⋅41​=0⋅41​, which gives us x=0x = 0x=0.One solution!Awesome!

STEP 10

Time for equation C: 2x+31=2x−312x + 31 = 2x - 312x+31=2x−31.

STEP 11

Let's **subtract** 2x2x2x from both sides: 2x+31−2x=2x−31−2x2x + 31 - 2x = 2x - 31 - 2x2x+31−2x=2x−31−2x, which simplifies to 31=−3131 = -3131=−31.Uh oh!That's not right!

STEP 12

Since 31 is *never* equal to -31, this equation has **no** solutions.Tricky!

STEP 13

Last but not least, equation D: 2x−2=2x−312x - 2 = 2x - 312x−2=2x−31.

STEP 14

Let's **subtract** 2x2x2x from both sides: 2x−2−2x=2x−31−2x2x - 2 - 2x = 2x - 31 - 2x2x−2−2x=2x−31−2x, which simplifies to −2=−31-2 = -31−2=−31.Nope, that's not true either!

STEP 15

So, equation D also has **no** solutions.

STEP 16

Only equation B, 2x−31=−2x−312x - 31 = -2x - 312x−31=−2x−31, has exactly **one** solution, which is x=0x = 0x=0.

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Studdy solves anything!

Start learning now

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QuestionWhich of the following equations hav exactly one solution?
Choose all answers that apply: A $2 x-31=2 x-31$ B $2 x-31=-2 x-31$ (c) $2 x+31=2 x-31$ (D) $2 x-2=2 x-31$

What is this asking? Which of these equations have only one solution for $x$ ? Watch out! Some equations can have no solutions or infinitely many solutions!
Don't fall for that trap!

1. Analyze Equation A
2. Analyze Equation B
3. Analyze Equation C
4. Analyze Equation D

Alright, let's start with equation A: $2x - 31 = 2x - 31$ .
This looks interesting!

Notice that both sides of the equation are exactly the same.
This means no matter what value we plug in for $x$ , the equation will always be true!

For example, if $x = 0$ , we get $-31 = -31$ .
If $x = 1$ , we get $-29 = -29$ .
See? Always true!
So, equation A has infinitely many solutions, not just one.

Now, let's move on to equation B: $2x - 31 = -2x - 31$ .

Let's add $2x$ to both sides to group the $x$ terms: $2x - 31 + 2x = -2x - 31 + 2x$ , which simplifies to $4x - 31 = -31$ .

Next, let's add 31 to both sides to isolate the $x$ term: $4x - 31 + 31 = -31 + 31$ , which simplifies to $4x = 0$ .

Finally, let's divide both sides by 4 to solve for $x$ : $4x \cdot \frac{1}{4} = 0 \cdot \frac{1}{4}$ , which gives us $x = 0$ .
One solution!
Awesome!

Time for equation C: $2x + 31 = 2x - 31$ .

Let's subtract $2x$ from both sides: $2x + 31 - 2x = 2x - 31 - 2x$ , which simplifies to $31 = -31$ .
Uh oh!
That's not right!

Since 31 is never equal to -31, this equation has no solutions.
Tricky!

Last but not least, equation D: $2x - 2 = 2x - 31$ .

Let's subtract $2x$ from both sides: $2x - 2 - 2x = 2x - 31 - 2x$ , which simplifies to $-2 = -31$ .
Nope, that's not true either!

So, equation D also has no solutions.

Only equation B, $2x - 31 = -2x - 31$ , has exactly one solution, which is $x = 0$ .