Solve the system of equations by elimination. $\newline$ $-2x - 3y + 3z = -19$ $\newline$ $3x + y + 2z = -16$ $\newline$ $2x - y + 2z = -10$

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Algebra 2

Solve a system of equations in three variables using elimination

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Q. Solve the system of equations by elimination.

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-2x - 3y + 3z = -19

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3x + y + 2z = -16

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2x - y + 2z = -10

Add Equations to Eliminate y: First, let's add the second and third equations to eliminate y. $\newline$ $(3x + y + 2z) + (2x - y + 2z) = -16 + (-10)$ $\newline$ $5x + 4z = -26$
Multiply Equations to Eliminate y: Now, let's multiply the first equation by $3$ and the second equation by $2$ so we can eliminate y again. $\newline$ $(-2x - 3y + 3z) \times 3 = -19 \times 3$ $\newline$ $(3x + y + 2z) \times 2 = -16 \times 2$ $\newline$ $-6x - 9y + 9z = -57$ $\newline$ $6x + 2y + 4z = -32$
Add Equations to Eliminate y: Next, we add the new equations from the previous step to eliminate y. $\newline$ $(-6x - 9y + 9z) + (6x + 2y + 4z) = -57 + (-32)$ $\newline$ $-7y + 13z = -89$
Align Coefficients of $z$ : Now we have two equations: $\newline$ $5x + 4z = -26$ $\newline$ $-7y + 13z = -89$ $\newline$ We need to find one more equation to eliminate another variable.
Add Equations to Eliminate z: Let's multiply the first original equation by $2$ and the third original equation by $3$ to align the coefficients of z. $(-2x - 3y + 3z) \times 2 = -19 \times 2$ $(2x - y + 2z) \times 3 = -10 \times 3$ $-4x - 6y + 6z = -38$ $6x - 3y + 6z = -30$
Solve for x: Now, we add these two new equations to eliminate z. $\newline$ $(-4x - 6y + 6z) + (6x - 3y + 6z) = -38 + (-30)$ $\newline$ $2x - 9y = -68$
Substitute $x$ into Equation: We now have three equations with two variables each: $5x + 4z = -26$ $-7y + 13z = -89$ $2x - 9y = -68$ We can solve one of these equations for a variable and substitute it into another equation.
Multiply to Remove Fraction: Let's solve the third equation for $x$ . $2x - 9y = -68$ $2x = 9y - 68$ $x = \frac{9y - 68}{2}$
Add Equations to Eliminate $y$ : Substitute $x$ into the first equation we found, $5x + 4z = -26$ . $\newline$ $5\left(\frac{9y - 68}{2}\right) + 4z = -26$ $\newline$ $\frac{45y - 340}{2} + 4z = -26$
Add Equations to Eliminate $y$ : Substitute $x$ into the first equation we found, $5x + 4z = -26$ .
$5\left(\frac{9y - 68}{2}\right) + 4z = -26$
$\frac{45y - 340}{2} + 4z = -26$ Multiply everything by $2$ to get rid of the fraction.
$45y - 340 + 8z = -52$
Add Equations to Eliminate $y$ : Substitute $x$ into the first equation we found, $5x + 4z = -26$ .
$5\left(\frac{9y - 68}{2}\right) + 4z = -26$
$\frac{45y - 340}{2} + 4z = -26$ Multiply everything by $2$ to get rid of the fraction.
$45y - 340 + 8z = -52$ Now, let's add this equation to the second equation we found, $-7y + 13z = -89$ , to eliminate $y$ .
$(45y - 340 + 8z) + (-7y + 13z) = -52 + (-89)$
$x$ $0$