Solve x + 2y + 2z = 3; 2x + y + 3z = - 1; 3x + 2y + z = 7 | Equatix - Math Solver

Fractions & Decimals

Ratios & Proportions

Basic Number Theory

Integers & Absolute Values

Factors & Multiples

Prime Factorization

Basic Equations & Inequalities

Coordinate Plane & Graphing

Linear Equations & Inequalities

Quadratic Equations

Polynomials & Factoring

Functions & Graphs

Systems of Equations

Trigonometric Functions

Trigonometric Identities & Equations

Right Triangle Trigonometry

Graphing Trigonometric Functions

Limits & Continuity

Derivatives & Applications

Integrals & Applications

Series & Sequences

Multivariable Calculus

Functions & Graphs

Polynomial & Rational Functions

Exponential & Logarithmic Functions

Sequences & Series

Descriptive Statistics

Normal Distribution

Hypothesis Testing

Regression Analysis

Linear Programming

Sets & Counting

Eigenvalues & Eigenvectors

Linear Transformations

Atomic Structure

Thermochemistry

Example

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

a⋅(b-2)=3b

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

a⋅(b-2)=3b

Question

$$x+2y+2z=3; 2x+y+3z=-1; 3x+2y+z=7$$

Solve for x, y, z

$x=1$
$y=3$
$z=-2$

Short Steps Using Substitution

Solve $x+2y+2z=3$ for $x$.

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

Substitute $-2y-2z+3$ for $x$ in the second and third equation.

$$2\left(-2y-2z+3\right)+y+3z=-1$$ $$3\left(-2y-2z+3\right)+2y+z=7$$

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

$$y=-\frac{1}{3}z+\frac{7}{3}$$ $$z=\frac{2}{5}-\frac{4}{5}y$$

Substitute $-\frac{1}{3}z+\frac{7}{3}$ for $y$ in the equation $z=\frac{2}{5}-\frac{4}{5}y$.

$$z=\frac{2}{5}-\frac{4}{5}\left(-\frac{1}{3}z+\frac{7}{3}\right)$$

Solve $z=\frac{2}{5}-\frac{4}{5}\left(-\frac{1}{3}z+\frac{7}{3}\right)$ for $z$.

$$z=-2$$

Substitute $-2$ for $z$ in the equation $y=-\frac{1}{3}z+\frac{7}{3}$.

$$y=-\frac{1}{3}\left(-2\right)+\frac{7}{3}$$

Calculate $y$ from $y=-\frac{1}{3}\left(-2\right)+\frac{7}{3}$.

$$y=3$$

Substitute $3$ for $y$ and $-2$ for $z$ in the equation $x=-2y-2z+3$.

$$x=-2\times 3-2\left(-2\right)+3$$

Calculate $x$ from $x=-2\times 3-2\left(-2\right)+3$.

$$x=1$$

The system is now solved.

$$x=1$$ $$y=3$$ $$z=-2$$