Solve P = 2x + 3y - 4z - 1; Q = 2y + 3x - 4z + 1; 7w + 3w - 47 - Q = 2y + 3x - 42 + 1 | 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

$$P=2x+3y-4z-1; Q=2y+3x-4z+1; 7w+3w-47-1Q=2y+3x-42+1$$

Solve for x, y, z

$x=\frac{Q}{5}-\frac{2P}{5}+2w-2$
$y=\frac{3P}{5}-\frac{4Q}{5}+2w$
$z=\frac{5w}{2}-\frac{Q}{2}-\frac{5}{4}$

Short Steps Using Substitution

Solve $P=2x+3y-4z-1$ for $x$.

$$x=\frac{1}{2}+2z-\frac{3}{2}y+\frac{1}{2}P$$

Substitute $\frac{1}{2}+2z-\frac{3}{2}y+\frac{1}{2}P$ for $x$ in the second and third equation.

$$Q=2y+3\left(\frac{1}{2}+2z-\frac{3}{2}y+\frac{1}{2}P\right)-4z+1$$ $$7w+3w-47-Q=2y+3\left(\frac{1}{2}+2z-\frac{3}{2}y+\frac{1}{2}P\right)-42+1$$

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

$$y=1+\frac{4}{5}z+\frac{3}{5}P-\frac{2}{5}Q$$ $$z=-\frac{5}{4}+\frac{5}{12}y-\frac{1}{4}P+\frac{5}{3}w-\frac{1}{6}Q$$

Substitute $1+\frac{4}{5}z+\frac{3}{5}P-\frac{2}{5}Q$ for $y$ in the equation $z=-\frac{5}{4}+\frac{5}{12}y-\frac{1}{4}P+\frac{5}{3}w-\frac{1}{6}Q$.

$$z=-\frac{5}{4}+\frac{5}{12}\left(1+\frac{4}{5}z+\frac{3}{5}P-\frac{2}{5}Q\right)-\frac{1}{4}P+\frac{5}{3}w-\frac{1}{6}Q$$

Solve $z=-\frac{5}{4}+\frac{5}{12}\left(1+\frac{4}{5}z+\frac{3}{5}P-\frac{2}{5}Q\right)-\frac{1}{4}P+\frac{5}{3}w-\frac{1}{6}Q$ for $z$.

$$z=-\frac{5}{4}-\frac{1}{2}Q+\frac{5}{2}w$$

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

$$y=1+\frac{4}{5}\left(-\frac{5}{4}-\frac{1}{2}Q+\frac{5}{2}w\right)+\frac{3}{5}P-\frac{2}{5}Q$$

Calculate $y$ from $y=1+\frac{4}{5}\left(-\frac{5}{4}-\frac{1}{2}Q+\frac{5}{2}w\right)+\frac{3}{5}P-\frac{2}{5}Q$.

$$y=\frac{3}{5}P-\frac{4}{5}Q+2w$$

Substitute $\frac{3}{5}P-\frac{4}{5}Q+2w$ for $y$ and $-\frac{5}{4}-\frac{1}{2}Q+\frac{5}{2}w$ for $z$ in the equation $x=\frac{1}{2}+2z-\frac{3}{2}y+\frac{1}{2}P$.

$$x=\frac{1}{2}+2\left(-\frac{5}{4}-\frac{1}{2}Q+\frac{5}{2}w\right)-\frac{3}{2}\left(\frac{3}{5}P-\frac{4}{5}Q+2w\right)+\frac{1}{2}P$$

Calculate $x$ from $x=\frac{1}{2}+2\left(-\frac{5}{4}-\frac{1}{2}Q+\frac{5}{2}w\right)-\frac{3}{2}\left(\frac{3}{5}P-\frac{4}{5}Q+2w\right)+\frac{1}{2}P$.

$$x=-2-\frac{2}{5}P+\frac{1}{5}Q+2w$$

The system is now solved.

$$x=-2-\frac{2}{5}P+\frac{1}{5}Q+2w$$ $$y=\frac{3}{5}P-\frac{4}{5}Q+2w$$ $$z=-\frac{5}{4}-\frac{1}{2}Q+\frac{5}{2}w$$