Full solution

Q. Solve the system of equations by substitution.

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-x - 3y - 3z = -6

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-3x - 3y - 2z = -4

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y = -7

Write Known Information: First, let's write down what we know: $\newline$ Total tape needed = $8,000 \, \text{cm}$ $\newline$ Tape per roll = $2,000 \, \text{cm}$ $\newline$ Now we divide the total tape needed by the tape per roll to find the number of rolls. $\newline$ $8,000 \, \text{cm} \div 2,000 \, \text{cm}$ per roll
Calculate Number of Rolls: Doing the division gives us: $\newline$ $8,000\,\text{cm} \div 2,000\,\text{cm}$ per roll = $4$ rolls $\newline$ So, the electrician needs to order $4$ rolls of tape.
Substitute and Simplify Equations: We are given $y = -7$ . Let's substitute $y$ in the other two equations. $\newline$ First equation: $-x - 3(-7) - 3z = -6$ $\newline$ Second equation: $-3x - 3(-7) - 2z = -4$
Isolate x in Equations: Now, simplify the equations: $\newline$ First equation: $-x + 21 - 3z = -6$ $\newline$ Second equation: $-3x + 21 - 2z = -4$
Solve for x: Next, we'll isolate $x$ in both equations: $\newline$ First equation: $-x - 3z = -6 - 21$ $\newline$ Second equation: $-3x - 2z = -4 - 21$
Substitute $x$ into Second Equation: Simplify the equations after subtracting: $\newline$ First equation: $-x - 3z = -27$ $\newline$ Second equation: $-3x - 2z = -25$
Distribute and Simplify: Now, let's solve for $x$ in the first equation: $\newline$ $-x = -27 + 3z$ $\newline$ $x = 27 - 3z$
Combine Like Terms: Substitute $x = 27 - 3z$ into the second equation: $\newline$ $-3(27 - 3z) - 2z = -25$
Solve for $z$ : Distribute and simplify: $-81 + 9z - 2z = -25$
Find $x$ : Combine like terms: $7z = 56$
Find $x$ : Combine like terms: $\newline$ $7z = 56$ Divide by $7$ to solve for $z$ : $\newline$ $z = 56 \div 7$ $\newline$ $z = 8$
Find $x$ : Combine like terms: $\newline$ $7z = 56$ Divide by $7$ to solve for $z$ : $\newline$ $z = \frac{56}{7}$ $\newline$ $z = 8$ Now we have $z = 8$ , let's find $x$ using $x = 27 - 3z$ : $\newline$ $x = 27 - 3(8)$ $\newline$ $7z = 56$ $0$ $\newline$ $7z = 56$ $1$