Solve x + y + z = 163; 12x + 13y + 14z = 221x; 13x + 14.5y + 15.5z = 237 | 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

$$x+y+z=163; 12x+13y+14z=221x; 13x+14.5y+15.5z=237$$

Solve for x, y, z

$x=-\frac{5}{147}\approx -0.034013605$
$y = \frac{336569}{147} = 2289\frac{86}{147} \approx 2289.585034014$
$z = -\frac{104201}{49} = -2126\frac{27}{49} \approx -2126.551020408$

Short Steps Using Substitution

Solve $x+y+z=163$ for $x$.

$$x=-y-z+163$$

Substitute $-y-z+163$ for $x$ in the second and third equation.

$$12\left(-y-z+163\right)+13y+14z=221\left(-y-z+163\right)$$ $$13\left(-y-z+163\right)+14.5y+15.5z=237$$

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

$$y=-\frac{223}{222}z+\frac{34067}{222}$$ $$z=-0.6y-752.8$$

Substitute $-\frac{223}{222}z+\frac{34067}{222}$ for $y$ in the equation $z=-0.6y-752.8$.

$$z=-0.6\left(-\frac{223}{222}z+\frac{34067}{222}\right)-752.8$$

Solve $z=-0.6\left(-\frac{223}{222}z+\frac{34067}{222}\right)-752.8$ for $z$.

$$z=-\frac{104201}{49}$$

Substitute $-\frac{104201}{49}$ for $z$ in the equation $y=-\frac{223}{222}z+\frac{34067}{222}$.

$$y=-\frac{223}{222}\left(-\frac{104201}{49}\right)+\frac{34067}{222}$$

Calculate $y$ from $y=-\frac{223}{222}\left(-\frac{104201}{49}\right)+\frac{34067}{222}$.

$$y=\frac{336569}{147}$$

Substitute $\frac{336569}{147}$ for $y$ and $-\frac{104201}{49}$ for $z$ in the equation $x=-y-z+163$.

$$x=-\frac{336569}{147}-\left(-\frac{104201}{49}\right)+163$$

Calculate $x$ from $x=-\frac{336569}{147}-\left(-\frac{104201}{49}\right)+163$.

$$x=-\frac{5}{147}$$

The system is now solved.

$$x=-\frac{5}{147}$$ $$y=\frac{336569}{147}$$ $$z=-\frac{104201}{49}$$