Question Find the number of solutions for the system of linear equations: $x+y+z=66$ and $5x+10y+20z=730$ , where $x, y, z$ are integers.

Studdy Solution

STEP 1

Assumptions
1. Let $x$ represent the number of five-dollar bills.
2. Let $y$ represent the number of ten-dollar bills.
3. Let $z$ represent the number of twenty-dollar bills.
4. The total number of bills is 66.
5. The total value of the bills is $730.

STEP 2

Translate the given information about the total number of bills into an equation.
$x + y + z = 66$

STEP 3

Translate the given information about the total value of the bills into an equation.
$5x + 10y + 20z = 730$

STEP 4

We now have a system of two equations with three variables:
$\begin{cases} x + y + z = 66 \\ 5x + 10y + 20z = 730 \end{cases}$

STEP 5

To determine the number of solutions, we need to analyze the system of equations. Since we have more variables (three) than equations (two), we expect an infinite number of solutions, assuming the equations are consistent and dependent.

STEP 6

Check if the two equations are consistent and dependent. To do this, we can compare the coefficients of the variables in both equations. If the second equation is a multiple of the first, then they are dependent.

STEP 7

Multiply the first equation by 5 to see if it matches the second equation:
$5(x + y + z) = 5 \cdot 66$

STEP 8

Distribute the 5 across the terms in the parentheses:
$5x + 5y + 5z = 330$

STEP 9

Compare the new equation with the second original equation:
$\begin{cases} 5x + 5y + 5z = 330 \\ 5x + 10y + 20z = 730 \end{cases}$

STEP 10

Observe that the new equation is not a multiple of the second original equation, as the coefficients of $y$ and $z$ are different. This means the two original equations are not dependent.

STEP 11

Since the two equations are not dependent, they represent two different constraints. However, with three variables and only two independent equations, there is not enough information to find a unique solution for each variable.

STEP 12

Conclude that the system of equations has an infinite number of solutions because there are more variables than independent equations.
There are a total of $\infty$ solutions.

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Question Find the number of solutions for the system of linear equations: $x+y+z=66$ and $5x+10y+20z=730$ , where $x, y, z$ are integers.

Studdy Solution

STEP 1

Assumptions
1. Let $x$ represent the number of five-dollar bills.
2. Let $y$ represent the number of ten-dollar bills.
3. Let $z$ represent the number of twenty-dollar bills.
4. The total number of bills is 66.
5. The total value of the bills is $730.

STEP 2

Translate the given information about the total number of bills into an equation.
$x + y + z = 66$

STEP 3

Translate the given information about the total value of the bills into an equation.
$5x + 10y + 20z = 730$

STEP 4

We now have a system of two equations with three variables:
$\begin{cases} x + y + z = 66 \\ 5x + 10y + 20z = 730 \end{cases}$

STEP 5

To determine the number of solutions, we need to analyze the system of equations. Since we have more variables (three) than equations (two), we expect an infinite number of solutions, assuming the equations are consistent and dependent.

STEP 6

Check if the two equations are consistent and dependent. To do this, we can compare the coefficients of the variables in both equations. If the second equation is a multiple of the first, then they are dependent.

STEP 7

Multiply the first equation by 5 to see if it matches the second equation:
$5(x + y + z) = 5 \cdot 66$

STEP 8

Distribute the 5 across the terms in the parentheses:
$5x + 5y + 5z = 330$

STEP 9

Compare the new equation with the second original equation:
$\begin{cases} 5x + 5y + 5z = 330 \\ 5x + 10y + 20z = 730 \end{cases}$

STEP 10

Observe that the new equation is not a multiple of the second original equation, as the coefficients of $y$ and $z$ are different. This means the two original equations are not dependent.

STEP 11

Since the two equations are not dependent, they represent two different constraints. However, with three variables and only two independent equations, there is not enough information to find a unique solution for each variable.

STEP 12

Conclude that the system of equations has an infinite number of solutions because there are more variables than independent equations.
There are a total of $\infty$ solutions.

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Question Find the number of solutions for the system of linear equations: x+y+z=66x+y+z=66x+y+z=66 and 5x+10y+20z=7305x+10y+20z=7305x+10y+20z=730, where x,y,zx, y, zx,y,z are integers.

Studdy Solution

STEP 1

Assumptions1. Let xxx represent the number of five-dollar bills.2. Let yyy represent the number of ten-dollar bills.3. Let zzz represent the number of twenty-dollar bills.4. The total number of bills is 66.5. The total value of the bills is $730.

STEP 2

Translate the given information about the total number of bills into an equation.x+y+z=66x + y + z = 66x+y+z=66

STEP 3

Translate the given information about the total value of the bills into an equation.5x+10y+20z=7305x + 10y + 20z = 7305x+10y+20z=730

STEP 4

We now have a system of two equations with three variables:{x+y+z=665x+10y+20z=730\begin{cases} x + y + z = 66 \\ 5x + 10y + 20z = 730 \end{cases}{x+y+z=665x+10y+20z=730​

STEP 5

To determine the number of solutions, we need to analyze the system of equations. Since we have more variables (three) than equations (two), we expect an infinite number of solutions, assuming the equations are consistent and dependent.

STEP 6

Check if the two equations are consistent and dependent. To do this, we can compare the coefficients of the variables in both equations. If the second equation is a multiple of the first, then they are dependent.

STEP 7

Multiply the first equation by 5 to see if it matches the second equation:5(x+y+z)=5⋅665(x + y + z) = 5 \cdot 665(x+y+z)=5⋅66

STEP 8

Distribute the 5 across the terms in the parentheses:5x+5y+5z=3305x + 5y + 5z = 3305x+5y+5z=330

STEP 9

Compare the new equation with the second original equation:{5x+5y+5z=3305x+10y+20z=730\begin{cases} 5x + 5y + 5z = 330 \\ 5x + 10y + 20z = 730 \end{cases}{5x+5y+5z=3305x+10y+20z=730​

STEP 10

Observe that the new equation is not a multiple of the second original equation, as the coefficients of yyy and zzz are different. This means the two original equations are not dependent.

STEP 11

Since the two equations are not dependent, they represent two different constraints. However, with three variables and only two independent equations, there is not enough information to find a unique solution for each variable.

STEP 12

Conclude that the system of equations has an infinite number of solutions because there are more variables than independent equations.There are a total of ∞\infty∞ solutions.

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Question Find the number of solutions for the system of linear equations: $x+y+z=66$ and $5x+10y+20z=730$ , where $x, y, z$ are integers.

Assumptions
1. Let $x$ represent the number of five-dollar bills.
2. Let $y$ represent the number of ten-dollar bills.
3. Let $z$ represent the number of twenty-dollar bills.
4. The total number of bills is 66.
5. The total value of the bills is $730.

Translate the given information about the total number of bills into an equation.
$x + y + z = 66$

Translate the given information about the total value of the bills into an equation.
$5x + 10y + 20z = 730$

We now have a system of two equations with three variables:
$\begin{cases} x + y + z = 66 \\ 5x + 10y + 20z = 730 \end{cases}$

Multiply the first equation by 5 to see if it matches the second equation:
$5(x + y + z) = 5 \cdot 66$

Distribute the 5 across the terms in the parentheses:
$5x + 5y + 5z = 330$

Compare the new equation with the second original equation:
$\begin{cases} 5x + 5y + 5z = 330 \\ 5x + 10y + 20z = 730 \end{cases}$

Observe that the new equation is not a multiple of the second original equation, as the coefficients of $y$ and $z$ are different. This means the two original equations are not dependent.

Conclude that the system of equations has an infinite number of solutions because there are more variables than independent equations.
There are a total of $\infty$ solutions.