sketch the graph or use a graphing utility to determine which of the following solution scenarios is true for the system of equations.
$$
\left\{\begin{array} { l } { 3 x + 7 y - z = 8 } \\ { x + y + z = 10 } \\ { x - y + 2 z = 3 } \end{array}\
ight.
$$
(1 point)
all three planes overlap, which means there are infinite solutions.
all three planes are parallel, which means there are no solutions.
all three planes intersect at a single point, which means there is one solution.
two of the planes are parallel, which means there are no solutions.

Question

Accepted Answer

# Explanation:
## Step1: Eliminate $z$ from the first two equations
Add $3x + 7y−z = 8$ and $x + y+z = 10$.
$$
\begin{align*}
(3x + 7y−z)+(x + y+z)&=8 + 10\
4x+8y&=18\
2x + 4y&=9
\end{align*}
$$
## Step2: Eliminate $z$ from the second and third equations
Multiply $x + y+z = 10$ by $2$: $2x+2y + 2z=20$.
Subtract $x−y + 2z = 3$ from it:
$$
\begin{align*}
(2x+2y + 2z)-(x−y + 2z)&=20 - 3\
2x+2y+2z - x + y-2z&=17\
x+3y&=17\
x&=17 - 3y
\end{align*}
$$
## Step3: Substitute $x = 17 - 3y$ into $2x + 4y=9$
$$
\begin{align*}
2(17-3y)+4y&=9\
34-6y + 4y&=9\
-2y&=9 - 34\
-2y&=-25\
y&=\frac{25}{2}
\end{align*}
$$
## Step4: Find $x$
Substitute $y=\frac{25}{2}$ into $x = 17 - 3y$
$$
\begin{align*}
x&=17-3\times\frac{25}{2}\
x&=\frac{34 - 75}{2}\
x&=-\frac{41}{2}
\end{align*}
$$
## Step5: Find $z$
Substitute $x =-\frac{41}{2}$ and $y=\frac{25}{2}$ into $x + y+z = 10$
$$
\begin{align*}
-\frac{41}{2}+\frac{25}{2}+z&=10\
\frac{-41 + 25}{2}+z&=10\
\frac{-16}{2}+z&=10\
-8+z&=10\
z&=18
\end{align*}
$$

# Answer:
All three planes intersect at a single point, which means there is one solution.

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Step1: Eliminate \(z\) from the first two equations

Step2: Eliminate \(z\) from the second and third equations

Step3: Substitute \(x = 17 - 3y\) into \(2x + 4y=9\)

Step4: Find \(x\)

Step5: Find \(z\)

Answer: