How many solutions does the system of equations below have? 2x + 3y + 2z = -19 -2x - 2y + 2z = 5 -x - 2y - 3z = -15 Choices: (A)no solution (B)one solution (C)infinitely many solutions

Combine Equations to Eliminate x: Combine the first and second equations to eliminate x. (2x + 3y + 2z) + (-2x - 2y + 2z) = -19 + 5 3y - 2y + 2z + 2z = -14 y + 4z = -14Combine Equations to Eliminate x: Combine the second and third equations to eliminate x. (-2x - 2y + 2z) - (-x - 2y - 3z) = 5 - (-15) -2x + x - 2y + 2y + 2z + 3z = 5 + 15 -x + 5z = 20 x = 20 - 5zSubstitute x into First Equation: Substitute x = 20 - 5z into the first equation. 2(20 - 5z) + 3y + 2z = -19 40 - 10z + 3y + 2z = -19 3y - 8z = -59Align y Terms: Now we have two equations with two variables: y + 4z = -14 3y - 8z = -59 Multiply the first equation by 3 to align y terms. 3(y + 4z) = 3(-14) 3y + 12z = -42Eliminate y: Subtract the new equation from the second equation to eliminate y. (3y - 8z) - (3y + 12z) = -59 - (-42) -8z - 12z = -59 + 42 -20z = -17 z = -17 / -20 z = 17/20Find y: Substitute z = 17/20 into y + 4z = -14 to find y. y + 4(17/20) = -14 y + 68/20 = -14 y + 17/5 = -14 y = -14 - 17/5 y = -70/5 - 17/5 y = -87/5Find x: Substitute z = 17/20 and y = -87/5 into x = 20 - 5z to find x. x = 20 - 5(17/20) x = 20 - 17/4 x = 80/4 - 17/4 x = 63/4

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Math Problems

Algebra 2

Determine the number of solutions to a system of equations in three variables

Full solution

Q. How many solutions does the system of equations below have?

\newline

2x + 3y + 2z = -19

\newline

-2x - 2y + 2z = 5

\newline

-x - 2y - 3z = -15

\newline

Choices:

\newline

(A)no solution

\newline

(B)one solution

\newline

(C)infinitely many solutions

Combine Equations to Eliminate $x$ : Combine the first and second equations to eliminate $x$ . $\$2x + 3y + 2z$ + ( $-2$ x - $2$ y + $2$ z) = $-19$ + $5$ \) $3y - 2y + 2z + 2z = -14$ $y + 4z = -14$
Combine Equations to Eliminate $x$ : Combine the second and third equations to eliminate $x$ .$\newline$( $-2x - 2y + 2z$ ) - ( $-x - 2y - 3z$ ) = $5$ - ( $-15$ )$\newline$ $-2$ x + x - $2$ y + $2$ y + $2$ z + $3$ z = $5 + 15$ $\newline$-x + $5$ z = $x$ $0$ $\newline$ $x$ $2$
Substitute $x$ into First Equation: Substitute $x = 20 - 5z$ into the first equation. $\newline$ $2(20 - 5z) + 3y + 2z = -19$ $\newline$ $40 - 10z + 3y + 2z = -19$ $\newline$ $3y - 8z = -59$
Align y Terms: Now we have two equations with two variables: $\newline$ $y + 4z = -14$ $\newline$ $3y - 8z = -59$ $\newline$ Multiply the first equation by $3$ to align y terms. $\newline$ $3(y + 4z) = 3(-14)$ $\newline$ $3y + 12z = -42$
Eliminate y: Subtract the new equation from the second equation to eliminate y. $\newline$ $(3y - 8z) - (3y + 12z) = -59 - (-42)$ $\newline$ $-8z - 12z = -59 + 42$ $\newline$ $-20z = -17$ $\newline$ $z = -17 / -20$ $\newline$ $z = 17/20$
Find $y$ : Substitute $z = \frac{17}{20}$ into $y + 4z = -14$ to find $y$ .
$y + 4\left(\frac{17}{20}\right) = -14$
$y + \frac{68}{20} = -14$
$y + \frac{17}{5} = -14$
$y = -14 - \frac{17}{5}$
$y = -\frac{70}{5} - \frac{17}{5}$
$y = -\frac{87}{5}$
Find $x$ : Substitute $z = \frac{17}{20}$ and $y = -\frac{87}{5}$ into $x = 20 - 5z$ to find $x$ .
$x = 20 - 5\left(\frac{17}{20}\right)$
$x = 20 - \frac{17}{4}$
$x = \frac{80}{4} - \frac{17}{4}$
$x = \frac{63}{4}$