QuestionFind the values of $h$ for which the matrix is consistent: $\begin{bmatrix} 1 & h & 4 \\ -3 & 6 & -16 \end{bmatrix}$ Choose A, B, C, or D.

Studdy Solution

STEP 1

Assumptions1. The given matrix is an augmented matrix of a linear system. . We are looking for the value(s) of $h$ that make the system consistent.
3. A consistent system is one that has at least one solution.

STEP 2

The given matrix represents the following system of linear equations $\begin{align} x + hy &=4 \\ -x +6y &= -16\end{align}$

STEP 3

We can multiply the first equation by3 to make the coefficients of $x$ in both equations the same. This will allow us to eliminate $x$ by adding the two equations together.
$\begin{align} 3x +3hy &=12 \\ -3x +6y &= -16\end{align}$

STEP 4

Now, we add the two equations together to eliminate $x$ $3hy +6y =12 -16$

STEP 5

implify the equation $3hy +y = -4$

STEP 6

Factor out $y$ from the left side of the equation $y(3h +6) = -4$

STEP 7

For the system to be consistent, the equation must have a solution for $y$ . This is only possible if $3h +6 \neq0$ , because if $3h +6 =0$ , we would be dividing by zero, which is undefined.

STEP 8

olve the inequality $3h +6 \neq0$ for $h$ $3h \neq -6$

STEP 9

Divide both sides by3 $h \neq -2$ So, the matrix is the augmented matrix of a consistent linear system if $h \neq -2$ . Therefore, the correct choice is B. The matrix is the augmented matrix of a consistent linear system if $h \neq -2$ .

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QuestionFind the values of $h$ for which the matrix is consistent: $\begin{bmatrix} 1 & h & 4 \\ -3 & 6 & -16 \end{bmatrix}$ Choose A, B, C, or D.

Studdy Solution

STEP 1

Assumptions1. The given matrix is an augmented matrix of a linear system. . We are looking for the value(s) of $h$ that make the system consistent.
3. A consistent system is one that has at least one solution.

STEP 2

The given matrix represents the following system of linear equations $\begin{align} x + hy &=4 \\ -x +6y &= -16\end{align}$

STEP 3

We can multiply the first equation by3 to make the coefficients of $x$ in both equations the same. This will allow us to eliminate $x$ by adding the two equations together.
$\begin{align} 3x +3hy &=12 \\ -3x +6y &= -16\end{align}$

STEP 4

Now, we add the two equations together to eliminate $x$ $3hy +6y =12 -16$

STEP 5

implify the equation $3hy +y = -4$

STEP 6

Factor out $y$ from the left side of the equation $y(3h +6) = -4$

STEP 7

For the system to be consistent, the equation must have a solution for $y$ . This is only possible if $3h +6 \neq0$ , because if $3h +6 =0$ , we would be dividing by zero, which is undefined.

STEP 8

olve the inequality $3h +6 \neq0$ for $h$ $3h \neq -6$

STEP 9

Divide both sides by3 $h \neq -2$ So, the matrix is the augmented matrix of a consistent linear system if $h \neq -2$ . Therefore, the correct choice is B. The matrix is the augmented matrix of a consistent linear system if $h \neq -2$ .

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QuestionFind the values of hhh for which the matrix is consistent: [1h4−36−16] \begin{bmatrix} 1 & h & 4 \\ -3 & 6 & -16 \end{bmatrix} [1−3​h6​4−16​] Choose A, B, C, or D.

Studdy Solution

STEP 1

Assumptions1. The given matrix is an augmented matrix of a linear system. . We are looking for the value(s) of hhh that make the system consistent.3. A consistent system is one that has at least one solution.

STEP 2

The given matrix represents the following system of linear equationsx+hy=4−x+6y=−16\begin{align*} x + hy &=4 \\ -x +6y &= -16\end{align*} x+hy−x+6y​=4=−16​

STEP 3

We can multiply the first equation by3 to make the coefficients of xxx in both equations the same. This will allow us to eliminate xxx by adding the two equations together.3x+3hy=12−3x+6y=−16\begin{align*} 3x +3hy &=12 \\ -3x +6y &= -16\end{align*} 3x+3hy−3x+6y​=12=−16​

STEP 4

Now, we add the two equations together to eliminate xxx3hy+6y=12−163hy +6y =12 -163hy+6y=12−16

STEP 5

implify the equation3hy+y=−43hy +y = -43hy+y=−4

STEP 6

Factor out yyy from the left side of the equationy(3h+6)=−4y(3h +6) = -4y(3h+6)=−4

STEP 7

For the system to be consistent, the equation must have a solution for yyy. This is only possible if 3h+6≠03h +6 \neq03h+6=0, because if 3h+6=03h +6 =03h+6=0, we would be dividing by zero, which is undefined.

STEP 8

olve the inequality 3h+6≠03h +6 \neq03h+6=0 for hhh3h≠−63h \neq -63h=−6

STEP 9

Divide both sides by3h≠−2h \neq -2h=−2So, the matrix is the augmented matrix of a consistent linear system if h≠−2h \neq -2h=−2. Therefore, the correct choice is B. The matrix is the augmented matrix of a consistent linear system if h≠−2h \neq -2h=−2.

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QuestionFind the values of $h$ for which the matrix is consistent: $\begin{bmatrix} 1 & h & 4 \\ -3 & 6 & -16 \end{bmatrix}$ Choose A, B, C, or D.

Assumptions1. The given matrix is an augmented matrix of a linear system. . We are looking for the value(s) of $h$ that make the system consistent.
3. A consistent system is one that has at least one solution.

The given matrix represents the following system of linear equations $\begin{align} x + hy &=4 \\ -x +6y &= -16\end{align}$

We can multiply the first equation by3 to make the coefficients of $x$ in both equations the same. This will allow us to eliminate $x$ by adding the two equations together.
$\begin{align} 3x +3hy &=12 \\ -3x +6y &= -16\end{align}$

Now, we add the two equations together to eliminate $x$ $3hy +6y =12 -16$

implify the equation $3hy +y = -4$

Factor out $y$ from the left side of the equation $y(3h +6) = -4$

For the system to be consistent, the equation must have a solution for $y$ . This is only possible if $3h +6 \neq0$ , because if $3h +6 =0$ , we would be dividing by zero, which is undefined.

olve the inequality $3h +6 \neq0$ for $h$ $3h \neq -6$

Divide both sides by3 $h \neq -2$ So, the matrix is the augmented matrix of a consistent linear system if $h \neq -2$ . Therefore, the correct choice is B. The matrix is the augmented matrix of a consistent linear system if $h \neq -2$ .