Solve the system of equations by substitution. $\newline$ $3x + y - 2z = 3$ $\newline$ $3x - 2y - z = 20$ $\newline$ $z = -7$

View step-by-step help

Home

Math Problems

Precalculus

Solve a system of equations in three variables using substitution

Full solution

Q. Solve the system of equations by substitution.

\newline

3x + y - 2z = 3

\newline

3x - 2y - z = 20

\newline

z = -7

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 Values: We already know that $z = -7$ , so we can substitute this value into the other two equations.
Solve for $y$ : Substitute $z = -7$ into the first equation: $3x + y - 2(-7) = 3$ . This simplifies to $3x + y + 14 = 3$ .
Substitute into Second Equation: Now, let's solve for $y$ : $y = 3 - 3x - 14$ . Simplify to get $y = -11 - 3x$ .
Combine Like Terms: Next, substitute $z = -7$ into the second equation: $3x - 2y - (-7) = 20$ . This simplifies to $3x - 2y + 7 = 20$ .
Solve for x: Now, let's substitute $y = -11 - 3x$ into the second equation: $3x - 2(-11 - 3x) + 7 = 20$ . $\newline$ This simplifies to $3x + 22 + 6x + 7 = 20$ .
Find $y$ : Combine like terms: $9x + 29 = 20$ .
Final Solution: Now, solve for $x$ : $9x = 20 - 29$ . $\newline$ This simplifies to $9x = -9$ .
Final Solution: Now, solve for $x$ : $9x = 20 - 29$ . $\newline$ This simplifies to $9x = -9$ .Divide both sides by $9$ to find $x$ : $x = -9 / 9$ . $\newline$ This simplifies to $x = -1$ .
Final Solution: Now, solve for $x$ : $9x = 20 - 29$ . This simplifies to $9x = -9$ .Divide both sides by $9$ to find $x$ : $x = -9 / 9$ . This simplifies to $x = -1$ .Now that we have $x$ , we can find $y$ by substituting $x$ back into the equation for $y$ : $9x = 20 - 29$ $1$ . This simplifies to $9x = 20 - 29$ $2$ .
Final Solution: Now, solve for $x$ : $9x = 20 - 29$ . This simplifies to $9x = -9$ . Divide both sides by $9$ to find $x$ : $x = -9 / 9$ . This simplifies to $x = -1$ . Now that we have $x$ , we can find $y$ by substituting $x$ back into the equation for $y$ : $9x = 20 - 29$ $1$ . This simplifies to $9x = 20 - 29$ $2$ . So, $9x = 20 - 29$ $3$ .
Final Solution: Now, solve for $x$ : $9x = 20 - 29$ . This simplifies to $9x = -9$ .Divide both sides by $9$ to find $x$ : $x = -9 / 9$ . This simplifies to $x = -1$ .Now that we have $x$ , we can find $y$ by substituting $x$ back into the equation for $y$ : $9x = 20 - 29$ $1$ . This simplifies to $9x = 20 - 29$ $2$ .So, $9x = 20 - 29$ $3$ .We have found $x = -1$ , $9x = 20 - 29$ $3$ , and we already know $9x = 20 - 29$ $6$ .