Solve the system of equations by substitution. $\newline$ $z = -8$ $\newline$ $3x + 2y - 3z = 5$ $\newline$ $-3x + y + z = -13$

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Precalculus

Solve a system of equations in three variables using substitution

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Q. Solve the system of equations by substitution.

\newline

z = -8

\newline

3x + 2y - 3z = 5

\newline

-3x + y + z = -13

Calculate Rolls Needed: First, let's find out how many rolls the electrician needs by dividing the total amount of tape needed by the amount of tape on each roll. $\newline$ $8,000 \, \text{cm} \div 2,000 \, \text{cm/roll} = 4 \, \text{rolls}$ .
Substitute $z$ in Equations: So, the electrician should order $4$ rolls of tape.
Solve for $3x + 2y$ : We already know $z = -8$ , let's substitute $z$ in the other equations.
Substitute $z$ in Third Equation: Substitute $z = -8$ into the second equation: $3x + 2y - 3(-8) = 5$ .
Solve for $-3x + y$ : Now, simplify the equation: $3x + 2y + 24 = 5$ .
Solve for y: Subtract $24$ from both sides to solve for $3x + 2y$ : $3x + 2y = 5 - 24$ .
Substitute $y$ in First Equation: This gives us $3x + 2y = -19$ .
Solve for x: Now, substitute $z = -8$ into the third equation: $-3x + y + (-8) = -13$ .
Substitute $x$ in $y$ equation: Simplify the equation: $-3x + y - 8 = -13$ .
Final Solution: Add $8$ to both sides to solve for $-3x + y$ : $-3x + y = -13 + 8$ .
Final Solution: Add $8$ to both sides to solve for $-3x + y$ : $-3x + y = -13 + 8$ .This gives us $-3x + y = -5$ .
Final Solution: Add $8$ to both sides to solve for $-3x + y$ : $-3x + y = -13 + 8$ . This gives us $-3x + y = -5$ . Now we have two equations with two variables: $3x + 2y = -19$ and $-3x + y = -5$ .
Final Solution: Add $8$ to both sides to solve for $-3x + y$ : $-3x + y = -13 + 8$ . This gives us $-3x + y = -5$ . Now we have two equations with two variables: $3x + 2y = -19$ and $-3x + y = -5$ . Let's solve the second equation for $y$ : $y = -5 + 3x$ .
Final Solution: Add $8$ to both sides to solve for $-3x + y$ : $-3x + y = -13 + 8$ . This gives us $-3x + y = -5$ . Now we have two equations with two variables: $3x + 2y = -19$ and $-3x + y = -5$ . Let's solve the second equation for $y$ : $y = -5 + 3x$ . Substitute $y = -5 + 3x$ into the first equation: $3x + 2(-5 + 3x) = -19$ .
Final Solution: Add $8$ to both sides to solve for $-3x + y$ : $-3x + y = -13 + 8$ . This gives us $-3x + y = -5$ . Now we have two equations with two variables: $3x + 2y = -19$ and $-3x + y = -5$ . Let's solve the second equation for $y$ : $y = -5 + 3x$ . Substitute $y = -5 + 3x$ into the first equation: $3x + 2(-5 + 3x) = -19$ . Expand the equation: $-3x + y$ $0$ .
Final Solution: Add $8$ to both sides to solve for $-3x + y$ : $-3x + y = -13 + 8$ . This gives us $-3x + y = -5$ . Now we have two equations with two variables: $3x + 2y = -19$ and $-3x + y = -5$ . Let's solve the second equation for $y$ : $y = -5 + 3x$ . Substitute $y = -5 + 3x$ into the first equation: $3x + 2(-5 + 3x) = -19$ . Expand the equation: $-3x + y$ $0$ . Combine like terms: $-3x + y$ $1$ .
Final Solution: Add $8$ to both sides to solve for $-3x + y$ : $-3x + y = -13 + 8$ . This gives us $-3x + y = -5$ . Now we have two equations with two variables: $3x + 2y = -19$ and $-3x + y = -5$ . Let's solve the second equation for $y$ : $y = -5 + 3x$ . Substitute $y = -5 + 3x$ into the first equation: $3x + 2(-5 + 3x) = -19$ . Expand the equation: $-3x + y$ $0$ . Combine like terms: $-3x + y$ $1$ . Add $-3x + y$ $2$ to both sides: $-3x + y$ $3$ .
Final Solution: Add $8$ to both sides to solve for $-3x + y$ : $-3x + y = -13 + 8$ . This gives us $-3x + y = -5$ . Now we have two equations with two variables: $3x + 2y = -19$ and $-3x + y = -5$ . Let's solve the second equation for $y$ : $y = -5 + 3x$ . Substitute $y = -5 + 3x$ into the first equation: $3x + 2(-5 + 3x) = -19$ . Expand the equation: $-3x + y$ $0$ . Combine like terms: $-3x + y$ $1$ . Add $-3x + y$ $2$ to both sides: $-3x + y$ $3$ . This gives us $-3x + y$ $4$ .
Final Solution: Add $8$ to both sides to solve for $-3x + y$ : $-3x + y = -13 + 8$ . This gives us $-3x + y = -5$ . Now we have two equations with two variables: $3x + 2y = -19$ and $-3x + y = -5$ . Let's solve the second equation for $y$ : $y = -5 + 3x$ . Substitute $y = -5 + 3x$ into the first equation: $3x + 2(-5 + 3x) = -19$ . Expand the equation: $-3x + y$ $0$ . Combine like terms: $-3x + y$ $1$ . Add $-3x + y$ $2$ to both sides: $-3x + y$ $3$ . This gives us $-3x + y$ $4$ . Divide both sides by $-3x + y$ $5$ to solve for $-3x + y$ $6$ : $-3x + y$ $7$ .
Final Solution: Add $8$ to both sides to solve for $-3x + y$ : $-3x + y = -13 + 8$ . This gives us $-3x + y = -5$ . Now we have two equations with two variables: $3x + 2y = -19$ and $-3x + y = -5$ . Let's solve the second equation for $y$ : $y = -5 + 3x$ . Substitute $y = -5 + 3x$ into the first equation: $3x + 2(-5 + 3x) = -19$ . Expand the equation: $-3x + y$ $0$ . Combine like terms: $-3x + y$ $1$ . Add $-3x + y$ $2$ to both sides: $-3x + y$ $3$ . This gives us $-3x + y$ $4$ . Divide both sides by $-3x + y$ $5$ to solve for $-3x + y$ $6$ : $-3x + y$ $7$ . This gives us $-3x + y$ $8$ .
Final Solution: Add $8$ to both sides to solve for $-3x + y$ : $-3x + y = -13 + 8$ . This gives us $-3x + y = -5$ . Now we have two equations with two variables: $3x + 2y = -19$ and $-3x + y = -5$ . Let's solve the second equation for $y$ : $y = -5 + 3x$ . Substitute $y = -5 + 3x$ into the first equation: $3x + 2(-5 + 3x) = -19$ . Expand the equation: $-3x + y$ $0$ . Combine like terms: $-3x + y$ $1$ . Add $-3x + y$ $2$ to both sides: $-3x + y$ $3$ . This gives us $-3x + y$ $4$ . Divide both sides by $-3x + y$ $5$ to solve for $-3x + y$ $6$ : $-3x + y$ $7$ . This gives us $-3x + y$ $8$ . Now substitute $-3x + y$ $8$ into $y = -5 + 3x$ : $-3x + y = -13 + 8$ $1$ .
Final Solution: Add $8$ to both sides to solve for $-3x + y$ : $-3x + y = -13 + 8$ . This gives us $-3x + y = -5$ . Now we have two equations with two variables: $3x + 2y = -19$ and $-3x + y = -5$ . Let's solve the second equation for $y$ : $y = -5 + 3x$ . Substitute $y = -5 + 3x$ into the first equation: $3x + 2(-5 + 3x) = -19$ . Expand the equation: $-3x + y$ $0$ . Combine like terms: $-3x + y$ $1$ . Add $-3x + y$ $2$ to both sides: $-3x + y$ $3$ . This gives us $-3x + y$ $4$ . Divide both sides by $-3x + y$ $5$ to solve for $-3x + y$ $6$ : $-3x + y$ $7$ . This gives us $-3x + y$ $8$ . Now substitute $-3x + y$ $8$ into $y = -5 + 3x$ : $-3x + y = -13 + 8$ $1$ . This simplifies to $-3x + y = -13 + 8$ $2$ .
Final Solution: Add $8$ to both sides to solve for $-3x + y$ : $-3x + y = -13 + 8$ . This gives us $-3x + y = -5$ . Now we have two equations with two variables: $3x + 2y = -19$ and $-3x + y = -5$ . Let's solve the second equation for $y$ : $y = -5 + 3x$ . Substitute $y = -5 + 3x$ into the first equation: $3x + 2(-5 + 3x) = -19$ . Expand the equation: $-3x + y$ $0$ . Combine like terms: $-3x + y$ $1$ . Add $-3x + y$ $2$ to both sides: $-3x + y$ $3$ . This gives us $-3x + y$ $4$ . Divide both sides by $-3x + y$ $5$ to solve for $-3x + y$ $6$ : $-3x + y$ $7$ . This gives us $-3x + y$ $8$ . Now substitute $-3x + y$ $8$ into $y = -5 + 3x$ : $-3x + y = -13 + 8$ $1$ . This simplifies to $-3x + y = -13 + 8$ $2$ . So, $-3x + y = -13 + 8$ $3$ .
Final Solution: Add $8$ to both sides to solve for $-3x + y$ : $-3x + y = -13 + 8$ . This gives us $-3x + y = -5$ . Now we have two equations with two variables: $3x + 2y = -19$ and $-3x + y = -5$ . Let's solve the second equation for $y$ : $y = -5 + 3x$ . Substitute $y = -5 + 3x$ into the first equation: $3x + 2(-5 + 3x) = -19$ . Expand the equation: $-3x + y$ $0$ . Combine like terms: $-3x + y$ $1$ . Add $-3x + y$ $2$ to both sides: $-3x + y$ $3$ . This gives us $-3x + y$ $4$ . Divide both sides by $-3x + y$ $5$ to solve for $-3x + y$ $6$ : $-3x + y$ $7$ . This gives us $-3x + y$ $8$ . Now substitute $-3x + y$ $8$ into $y = -5 + 3x$ : $-3x + y = -13 + 8$ $1$ . This simplifies to $-3x + y = -13 + 8$ $2$ . So, $-3x + y = -13 + 8$ $3$ . We have found $-3x + y$ $8$ , $-3x + y = -13 + 8$ $3$ , and we already know $-3x + y = -13 + 8$ $6$ .