Solve x + y + z = 6; x - y + 2Z = 5; 3x + y + z = 8 | Equatix - Math Solver

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Example

3(r+2s)=2t-4

a⋅(b-2)=3b

3(r+2s)=2t-4

a⋅(b-2)=3b

Question

$$x+y+z=6; x-y+2Z=5; 3x+y+z=8$$

Solve for x, y, z

$x=1$
$y=2\left(Z-2\right)$
$z=9-2Z$

Short Steps Using Substitution

Solve $x+y+z=6$ for $x$.

$$x=-y-z+6$$

Substitute $-y-z+6$ for $x$ in the second and third equation.

$$-y-z+6-y+2Z=5$$ $$3\left(-y-z+6\right)+y+z=8$$

Solve these equations for $y$ and $z$ respectively.

$$y=-\frac{1}{2}z+\frac{1}{2}+Z$$ $$z=5-y$$

Substitute $-\frac{1}{2}z+\frac{1}{2}+Z$ for $y$ in the equation $z=5-y$.

$$z=5-\left(-\frac{1}{2}z+\frac{1}{2}+Z\right)$$

Solve $z=5-\left(-\frac{1}{2}z+\frac{1}{2}+Z\right)$ for $z$.

$$z=9-2Z$$

Substitute $9-2Z$ for $z$ in the equation $y=-\frac{1}{2}z+\frac{1}{2}+Z$.

$$y=-\frac{1}{2}\left(9-2Z\right)+\frac{1}{2}+Z$$

Calculate $y$ from $y=-\frac{1}{2}\left(9-2Z\right)+\frac{1}{2}+Z$.

$$y=-4+2Z$$

Substitute $-4+2Z$ for $y$ and $9-2Z$ for $z$ in the equation $x=-y-z+6$.

$$x=-\left(-4+2Z\right)-\left(9-2Z\right)+6$$

Calculate $x$ from $x=-\left(-4+2Z\right)-\left(9-2Z\right)+6$.

$$x=1$$

The system is now solved.

$$x=1$$ $$y=-4+2Z$$ $$z=9-2Z$$