Solve the system of equations by substitution. $\newline$ $x - y - 2z = 19$ $\newline$ $z = -9$ $\newline$ $3x - 2y - z = 7$

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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

x - y - 2z = 19

\newline

z = -9

\newline

3x - 2y - z = 7

Calculate Rolls Needed: First, let's figure out how many rolls the electrician needs by dividing the total amount of tape needed by the amount of tape on each roll. $8,000 \, \text{cm} \div 2,000 \, \text{cm}/\text{roll} = 4 \, \text{rolls}.$
Substitute Value for $z$ : We already know that $z = -9$ , so we can substitute this value into the other two equations.
Solve for $x$ and $y$ : Substitute $z = -9$ into the first equation: $x - y - 2(-9) = 19$ . This simplifies to $x - y + 18 = 19$ .
Substitute Values into Equations: Now, let's solve for $x - y$ . Subtract $18$ from both sides: $x - y = 19 - 18$ . $\newline$ $x - y = 1$ .
Align Y Terms: Substitute $z = -9$ into the third equation: $3x - 2y - (-9) = 7$ . This simplifies to $3x - 2y + 9 = 7$ .
Solve for x: Now, let's solve for $3x - 2y$ . Subtract $9$ from both sides: $3x - 2y = 7 - 9$ . $\newline$ $3x - 2y = -2$ .
Solve for x: Now, let's solve for $3x - 2y$ . Subtract $9$ from both sides: $3x - 2y = 7 - 9$ . $\newline$ $3x - 2y = -2$ .We have two equations now: $x - y = 1$ and $3x - 2y = -2$ . $\newline$ Let's multiply the first equation by $2$ to align the $y$ terms: $2(x - y) = 2(1)$ . $\newline$ This gives us $2x - 2y = 2$ .
Solve for x: Now, let's solve for $3x - 2y$ . Subtract $9$ from both sides: $3x - 2y = 7 - 9$ . $3x - 2y = -2$ . We have two equations now: $x - y = 1$ and $3x - 2y = -2$ . Let's multiply the first equation by $2$ to align the y terms: $2(x - y) = 2(1)$ . This gives us $2x - 2y = 2$ . Now we have a system of two equations with two variables: $2x - 2y = 2$ $3x - 2y = -2$
Solve for x: Now, let's solve for $3x - 2y$ . Subtract $9$ from both sides: $3x - 2y = 7 - 9$ . $3x - 2y = -2$ .We have two equations now: $x - y = 1$ and $3x - 2y = -2$ . Let's multiply the first equation by $2$ to align the $y$ terms: $2(x - y) = 2(1)$ . This gives us $2x - 2y = 2$ .Now we have a system of two equations with two variables: $2x - 2y = 2$ $3x - 2y = -2$ Subtract the first equation from the second to solve for $9$ $2$ : $9$ $3$ . This simplifies to $9$ $4$ .