Math / Algebra

QuestionSort these equations into three types: true for all values, true for one value, or true for no values.
True for All Values $3(n+1)=3 n+1$ $2 n=2 n$
True for No Values True for One Value $7-r=r-7$ $v+2=v-2$ $y \cdot-6 \cdot-3=2 \cdot y \cdot 9$ $2 n=n$ $\frac{1}{2}+x=\frac{1}{3}+x$

Studdy Solution

STEP 1

1. We need to categorize each equation as being true for all values, true for one value, or true for no values.
2. An equation is true for all values if it holds for every possible value of the variable.
3. An equation is true for one value if there is exactly one solution that satisfies the equation.
4. An equation is true for no values if there are no solutions that satisfy the equation.

STEP 2

1. Analyze each equation to determine its type.
2. Categorize each equation based on the analysis.
3. 1. Categorize the equations.

STEP 3

Analyze the equation $3(n+1)=3n+1$ .
Distribute the left side: $3n + 3 = 3n + 1$ .
Simplify: $3 \neq 1$ .
This equation is true for no values.

STEP 4

Analyze the equation $2n=2n$ .
This is an identity, true for all values of $n$ .

STEP 5

Analyze the equation $7-r=r-7$ .
Rearrange: $7 = 2r - 7$ .
Solve for $r$ : $2r = 14$ , $r = 7$ .
This equation is true for one value.

STEP 6

Analyze the equation $v+2=v-2$ .
Simplify: $2 = -2$ .
This equation is true for no values.

STEP 7

Analyze the equation $y \cdot -6 \cdot -3=2 \cdot y \cdot 9$ .
Simplify both sides: $18y = 18y$ .
This is an identity, true for all values of $y$ .

STEP 8

Analyze the equation $2n=n$ .
Rearrange: $n = 0$ .
This equation is true for one value.

STEP 9

Analyze the equation $\frac{1}{2}+x=\frac{1}{3}+x$ .
Simplify: $\frac{1}{2} = \frac{1}{3}$ .
This equation is true for no values.

STEP 10

True for All Values: - $2n=2n$ - $y \cdot -6 \cdot -3=2 \cdot y \cdot 9$
True for One Value: - $7-r=r-7$ - $2n=n$
True for No Values: - $3(n+1)=3n+1$ - $v+2=v-2$ - $\frac{1}{2}+x=\frac{1}{3}+x$

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QuestionSort these equations into three types: true for all values, true for one value, or true for no values.
True for All Values $3(n+1)=3 n+1$ $2 n=2 n$
True for No Values True for One Value $7-r=r-7$ $v+2=v-2$ $y \cdot-6 \cdot-3=2 \cdot y \cdot 9$ $2 n=n$ $\frac{1}{2}+x=\frac{1}{3}+x$

Studdy Solution

STEP 1

STEP 2

1. Analyze each equation to determine its type.
2. Categorize each equation based on the analysis.
3. 1. Categorize the equations.

STEP 3

Analyze the equation $3(n+1)=3n+1$ .
Distribute the left side: $3n + 3 = 3n + 1$ .
Simplify: $3 \neq 1$ .
This equation is true for no values.

STEP 4

Analyze the equation $2n=2n$ .
This is an identity, true for all values of $n$ .

STEP 5

Analyze the equation $7-r=r-7$ .
Rearrange: $7 = 2r - 7$ .
Solve for $r$ : $2r = 14$ , $r = 7$ .
This equation is true for one value.

STEP 6

Analyze the equation $v+2=v-2$ .
Simplify: $2 = -2$ .
This equation is true for no values.

STEP 7

Analyze the equation $y \cdot -6 \cdot -3=2 \cdot y \cdot 9$ .
Simplify both sides: $18y = 18y$ .
This is an identity, true for all values of $y$ .

STEP 8

Analyze the equation $2n=n$ .
Rearrange: $n = 0$ .
This equation is true for one value.

STEP 9

Analyze the equation $\frac{1}{2}+x=\frac{1}{3}+x$ .
Simplify: $\frac{1}{2} = \frac{1}{3}$ .
This equation is true for no values.

STEP 10

True for All Values: - $2n=2n$ - $y \cdot -6 \cdot -3=2 \cdot y \cdot 9$
True for One Value: - $7-r=r-7$ - $2n=n$
True for No Values: - $3(n+1)=3n+1$ - $v+2=v-2$ - $\frac{1}{2}+x=\frac{1}{3}+x$

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Studdy Solution

STEP 1

STEP 2

1. Analyze each equation to determine its type.2. Categorize each equation based on the analysis.3. 1. Categorize the equations.

STEP 3

Analyze the equation 3(n+1)=3n+13(n+1)=3n+13(n+1)=3n+1.Distribute the left side: 3n+3=3n+13n + 3 = 3n + 13n+3=3n+1.Simplify: 3≠13 \neq 13=1.This equation is true for no values.

STEP 4

Analyze the equation 2n=2n2n=2n2n=2n.This is an identity, true for all values of nnn.

STEP 5

Analyze the equation 7−r=r−77-r=r-77−r=r−7.Rearrange: 7=2r−77 = 2r - 77=2r−7.Solve for rrr: 2r=142r = 142r=14, r=7r = 7r=7.This equation is true for one value.

STEP 6

Analyze the equation v+2=v−2v+2=v-2v+2=v−2.Simplify: 2=−22 = -22=−2.This equation is true for no values.

STEP 7

Analyze the equation y⋅−6⋅−3=2⋅y⋅9y \cdot -6 \cdot -3=2 \cdot y \cdot 9y⋅−6⋅−3=2⋅y⋅9.Simplify both sides: 18y=18y18y = 18y18y=18y.This is an identity, true for all values of yyy.

STEP 8

Analyze the equation 2n=n2n=n2n=n.Rearrange: n=0n = 0n=0.This equation is true for one value.

STEP 9

Analyze the equation 12+x=13+x\frac{1}{2}+x=\frac{1}{3}+x21​+x=31​+x.Simplify: 12=13\frac{1}{2} = \frac{1}{3}21​=31​.This equation is true for no values.

STEP 10

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1. Analyze each equation to determine its type.
2. Categorize each equation based on the analysis.
3. 1. Categorize the equations.

Analyze the equation $3(n+1)=3n+1$ .
Distribute the left side: $3n + 3 = 3n + 1$ .
Simplify: $3 \neq 1$ .
This equation is true for no values.

Analyze the equation $2n=2n$ .
This is an identity, true for all values of $n$ .

Analyze the equation $7-r=r-7$ .
Rearrange: $7 = 2r - 7$ .
Solve for $r$ : $2r = 14$ , $r = 7$ .
This equation is true for one value.

Analyze the equation $v+2=v-2$ .
Simplify: $2 = -2$ .
This equation is true for no values.

Analyze the equation $y \cdot -6 \cdot -3=2 \cdot y \cdot 9$ .
Simplify both sides: $18y = 18y$ .
This is an identity, true for all values of $y$ .

Analyze the equation $2n=n$ .
Rearrange: $n = 0$ .
This equation is true for one value.

Analyze the equation $\frac{1}{2}+x=\frac{1}{3}+x$ .
Simplify: $\frac{1}{2} = \frac{1}{3}$ .
This equation is true for no values.