QuestionA 10-foot board is cut into 3 pieces. One piece is 1 foot longer than the shortest and 2 feet shorter than the longest. Find the lengths.

Studdy Solution

STEP 1

Assumptions1. The total length of the board is10 feet. . The board is cut into three pieces.
3. One piece is one foot longer than the shortest piece.
4. The same piece is also two feet shorter than the longest piece.

STEP 2

Let's denote the lengths of the three pieces as $x$ , $y$ , and $z$ , where $x$ is the shortest piece, $y$ is the piece that is one foot longer than $x$ and two feet shorter than $z$ , and $z$ is the longest piece.

STEP 3

From the problem, we know that $y = x +1$ and $y = z -2$ . We can write these equations as $y = x +1$ $y = z -2$

STEP 4

We also know that the total length of the board, which is the sum of the lengths of the three pieces, is10 feet. So, we can write this as $x + y + z =10$

STEP 5

Now, we have a system of three equations. We can solve this system by substitution. First, substitute $y$ from the first equation into the third equation $x + (x +1) + z =10$

STEP 6

implify the equation $2x + z +1 =10$

STEP 7

Rearrange the equation to isolate $z$ $z =10 -2x -1$

STEP 8

implify the equation $z = -2x$

STEP 9

Now, substitute $y$ from the second equation into the equation we just found $z = (z -2) +2$

STEP 10

implify the equation $z = z$

STEP 11

This equation is always true, so it doesn't give us any new information. Instead, substitute $z$ from the equation $z =9 -x$ into the equation $y = z -$ $y = (9 -x) -$

STEP 12

implify the equation $y =7 -2x$

STEP 13

Now, we have two equations for $y$ : $y = x +$ and $y =7 -2x$ . Set these two equations equal to each other and solve for $x$ $x + =7 -2x$

STEP 14

Rearrange the equation to isolate $x$ $3x =6$

STEP 15

Divide both sides by3 to solve for $x$ $x =2$

STEP 16

Now that we have $x$ , we can find $y$ and $z$ by substituting $x$ into the equations $y = x +$ and $z =9 -2x$ $y =2 + =3$ $z =9 -2\\times2 =5$ So, the lengths of the three pieces are2 feet,3 feet, and5 feet.

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QuestionA 10-foot board is cut into 3 pieces. One piece is 1 foot longer than the shortest and 2 feet shorter than the longest. Find the lengths.

Studdy Solution

STEP 1

Assumptions1. The total length of the board is10 feet. . The board is cut into three pieces.
3. One piece is one foot longer than the shortest piece.
4. The same piece is also two feet shorter than the longest piece.

STEP 2

Let's denote the lengths of the three pieces as $x$ , $y$ , and $z$ , where $x$ is the shortest piece, $y$ is the piece that is one foot longer than $x$ and two feet shorter than $z$ , and $z$ is the longest piece.

STEP 3

From the problem, we know that $y = x +1$ and $y = z -2$ . We can write these equations as $y = x +1$ $y = z -2$

STEP 4

We also know that the total length of the board, which is the sum of the lengths of the three pieces, is10 feet. So, we can write this as $x + y + z =10$

STEP 5

Now, we have a system of three equations. We can solve this system by substitution. First, substitute $y$ from the first equation into the third equation $x + (x +1) + z =10$

STEP 6

implify the equation $2x + z +1 =10$

STEP 7

Rearrange the equation to isolate $z$ $z =10 -2x -1$

STEP 8

implify the equation $z = -2x$

STEP 9

Now, substitute $y$ from the second equation into the equation we just found $z = (z -2) +2$

STEP 10

implify the equation $z = z$

STEP 11

This equation is always true, so it doesn't give us any new information. Instead, substitute $z$ from the equation $z =9 -x$ into the equation $y = z -$ $y = (9 -x) -$

STEP 12

implify the equation $y =7 -2x$

STEP 13

Now, we have two equations for $y$ : $y = x +$ and $y =7 -2x$ . Set these two equations equal to each other and solve for $x$ $x + =7 -2x$

STEP 14

Rearrange the equation to isolate $x$ $3x =6$

STEP 15

Divide both sides by3 to solve for $x$ $x =2$

STEP 16

Now that we have $x$ , we can find $y$ and $z$ by substituting $x$ into the equations $y = x +$ and $z =9 -2x$ $y =2 + =3$ $z =9 -2\\times2 =5$ So, the lengths of the three pieces are2 feet,3 feet, and5 feet.

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QuestionA 10-foot board is cut into 3 pieces. One piece is 1 foot longer than the shortest and 2 feet shorter than the longest. Find the lengths.

Studdy Solution

STEP 1

Assumptions1. The total length of the board is10 feet. . The board is cut into three pieces.3. One piece is one foot longer than the shortest piece.4. The same piece is also two feet shorter than the longest piece.

STEP 2

Let's denote the lengths of the three pieces as xxx, yyy, and zzz, where xxx is the shortest piece, yyy is the piece that is one foot longer than xxx and two feet shorter than zzz, and zzz is the longest piece.

STEP 3

From the problem, we know that y=x+1y = x +1y=x+1 and y=z−2y = z -2y=z−2. We can write these equations asy=x+1y = x +1y=x+1y=z−2y = z -2y=z−2

STEP 4

We also know that the total length of the board, which is the sum of the lengths of the three pieces, is10 feet. So, we can write this asx+y+z=10x + y + z =10x+y+z=10

STEP 5

Now, we have a system of three equations. We can solve this system by substitution. First, substitute yyy from the first equation into the third equationx+(x+1)+z=10x + (x +1) + z =10x+(x+1)+z=10

STEP 6

implify the equation2x+z+1=102x + z +1 =102x+z+1=10

STEP 7

Rearrange the equation to isolate zzzz=10−2x−1z =10 -2x -1z=10−2x−1

STEP 8

implify the equationz=−2xz = -2xz=−2x

STEP 9

Now, substitute yyy from the second equation into the equation we just foundz=(z−2)+2z = (z -2) +2z=(z−2)+2

STEP 10

implify the equationz=zz = zz=z

STEP 11

This equation is always true, so it doesn't give us any new information. Instead, substitute zzz from the equation z=9−xz =9 -xz=9−x into the equation y=z−y = z -y=z−y=(9−x)−y = (9 -x) -y=(9−x)−

STEP 12

implify the equationy=7−2xy =7 -2xy=7−2x

STEP 13

Now, we have two equations for yyy: y=x+y = x +y=x+ and y=7−2xy =7 -2xy=7−2x. Set these two equations equal to each other and solve for xxxx+=7−2xx + =7 -2xx+=7−2x

STEP 14

Rearrange the equation to isolate xxx3x=63x =63x=6

STEP 15

Divide both sides by3 to solve for xxxx=2x =2x=2

STEP 16

Now that we have xxx, we can find yyy and zzz by substituting xxx into the equations y=x+y = x +y=x+ and z=9−2xz =9 -2xz=9−2xy=2+=3y =2 + =3y=2+=3z=9−2times2=5z =9 -2\\times2 =5z=9−2times2=5So, the lengths of the three pieces are2 feet,3 feet, and5 feet.

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Assumptions1. The total length of the board is10 feet. . The board is cut into three pieces.
3. One piece is one foot longer than the shortest piece.
4. The same piece is also two feet shorter than the longest piece.

Let's denote the lengths of the three pieces as $x$ , $y$ , and $z$ , where $x$ is the shortest piece, $y$ is the piece that is one foot longer than $x$ and two feet shorter than $z$ , and $z$ is the longest piece.

From the problem, we know that $y = x +1$ and $y = z -2$ . We can write these equations as $y = x +1$ $y = z -2$

We also know that the total length of the board, which is the sum of the lengths of the three pieces, is10 feet. So, we can write this as $x + y + z =10$

Now, we have a system of three equations. We can solve this system by substitution. First, substitute $y$ from the first equation into the third equation $x + (x +1) + z =10$

implify the equation $2x + z +1 =10$

Rearrange the equation to isolate $z$ $z =10 -2x -1$

implify the equation $z = -2x$

Now, substitute $y$ from the second equation into the equation we just found $z = (z -2) +2$

implify the equation $z = z$

This equation is always true, so it doesn't give us any new information. Instead, substitute $z$ from the equation $z =9 -x$ into the equation $y = z -$ $y = (9 -x) -$

implify the equation $y =7 -2x$

Now, we have two equations for $y$ : $y = x +$ and $y =7 -2x$ . Set these two equations equal to each other and solve for $x$ $x + =7 -2x$

Rearrange the equation to isolate $x$ $3x =6$

Divide both sides by3 to solve for $x$ $x =2$

Now that we have $x$ , we can find $y$ and $z$ by substituting $x$ into the equations $y = x +$ and $z =9 -2x$ $y =2 + =3$ $z =9 -2\\times2 =5$ So, the lengths of the three pieces are2 feet,3 feet, and5 feet.