plus( X_TERM, A+Y, B+Z, C ) = plus( D+Y, E+Z, F )
Solve for X
.
ANSWER( D-A, E-B, F-C, G )
ANSWER( D+A, E-B, F-C, G )
ANSWER( D-A, E+B, F-C, G )
ANSWER( D-A, E-B, F+C, G )
ANSWER( D+A, E+B, F-C, G )
ANSWER( D+A, E-B, F+C, G )
ANSWER( D-A, E+B, F+C, G )
ANSWER( D+A, E+B, F+C, G )
Combine constant terms on the right.
plus( X_TERM, A+Y, B+Z, color( C, true ) ) = plus( D+Y, E+Z, color( F, true ) )
plus( X_TERM, A+Y, B+Z ) = plus( D+Y, E+Z, color( F-C, true ) )
Combine Z
terms on the right.
plus( X_TERM, A+Y, color( B+Z, false ) ) = plus( D+Y, color( E+Z, false ), F-C )
plus( X_TERM, A+Y ) = plus( D+Y, color( (E-B)+Z, false ), F-C )
Combine Y
terms on the right.
plus( X_TERM, color( A+Y, true ) ) = plus( color( D+Y, true ), (E-B)+Z, F-C )
plus( X_TERM ) = plus( color( (D-A)+Y, true ), (E-B)+Z, F-C )
Isolate X
.
plus( color( G, false ) + X + color( X_EXTRAS, false ) ) = plus( (D-A)+Y, (E-B)+Z, F-C )
X = \dfrac{ plus( (D-A)+Y, (E-B)+Z, F-C ) }{ plus( color( G + X_EXTRAS, false ) ) }
All of these terms are divisible by GCD
.
Divide by the common factor and swap signs so the denominator isn't negative.
Divide by the common factor.
Swap the signs so the denominator isn't negative.
X = \dfrac{ plus( color( round( DIVISOR*(D-A) ), true )+Y, color( round( DIVISOR*(E-B) ), true )+Z, color( round( DIVISOR*(F-C) ), true ) ) }{ plus( color( round( DIVISOR*G )+X_EXTRAS, true ) ) }
plus( A+X+Y, B+X+Z, C+X, D ) = plus( E+Y, F )
Solve for X
.
ANSWER( E, F-D, A, B, C )
ANSWER( E, F+D, A, B, C )
ANSWER( 0, F-D, A, B, C )
ANSWER( E, 0, A, B, C )
ANSWER( E, F-D, 0, B, C )
ANSWER( E, F-D, A, 0, C )
ANSWER( E, F-D, A, B, 0 )
ANSWER( E+A, F-D, A, B, C )
ANSWER( E-A, F-D, A, B, C )
ANSWER( E, F-D, A+B, B, C )
ANSWER( E, F-D, A-B, B, C )
ANSWER( E, F-D, A, A+B, C )
ANSWER( E, F-D, A, A-B, C )
ANSWER( E, F-D, A, B, A+C )
ANSWER( E, F-D, A, B, A-C )
Combine constant terms on the right.
plus( A+X+Y, B+X+Z, C+X, color( D, true ) ) = plus( E+Y, color( F, true ) )
plus( A+X+Y, B+X+Z, C+X ) = plus( E+Y, color( F-D, true ) )
Notice that all the terms on the left-hand side of the equation have X
in them.
plus( A+color( X, false )+Y, B+color( X, false )+Z, C+color( X, false ) ) = plus( E+Y, F-D )
Factor out the X
.
color( X, false ) \cdot \left( plus( A+Y, B+Z, C ) \right) = plus( E+Y, F-D )
Isolate the X
.
X \cdot \left( color( plus( A+Y, B+Z, C ), true ) \right) = plus( E+Y, F-D )
X = \dfrac{ plus( E+Y, F-D ) }{ color( plus( A+Y, B+Z, C ), true ) }
We can simplify this by multiplying the top and bottom by -1
.
ANSWER( E, F-D, A, B, C )
\dfrac{ plus( A+X, B+Y ) }{ C } = \dfrac{ plus( D+X, E+Z ) }{ F }
Solve for X
.
ANSWER( E_TERM, -B_TERM, A_TERM-D_TERM )
ANSWER( E_TERM, B_TERM, A_TERM-D_TERM )
ANSWER( -E_TERM, -B_TERM, A_TERM-D_TERM )
ANSWER( -E_TERM, B_TERM, A_TERM-D_TERM )
ANSWER( E_TERM, -B_TERM, A_TERM+D_TERM )
ANSWER( E_TERM, B_TERM, A_TERM+D_TERM )
ANSWER( -E_TERM, -B_TERM, A_TERM+D_TERM )
ANSWER( -E_TERM, B_TERM, A_TERM+D_TERM )
ANSWER( E_TERM, -B_TERM, -A_TERM-D_TERM )
ANSWER( E_TERM, B_TERM, -A_TERM-D_TERM )
ANSWER( -E_TERM, B_TERM, -A_TERM-D_TERM )
ANSWER( -E_TERM, -B_TERM, -A_TERM-D_TERM )
Notice that the left- and right- denominators are the sameopposite.
\dfrac{ plus( A+X, B+Y ) }{ color( C, true ) } = \dfrac{ plus( D+X, E+Z ) }{ color( F, true ) }
So we can multiply both sides by C
.
color( C, true ) \cdot \dfrac{ plus( A+X, B+Y ) }{ color( C, true ) } = color( C, true ) \cdot \dfrac{ plus( D+X, E+Z ) }{ color( F, true ) }
plus( A+X, B+Y ) =
plus( D+X, E+Z )
color( "-", true ) \cdot \left( plus( D+X, E+Z ) \right)
Distribute the negative sign on the right side.
plus( A+X, B+Y ) = plus( D_TERM+X, E_TERM+Z )
plus( color( A_TERM, true )+X, color( B_TERM, true )+Y ) = plus( color( D_TERM, true )+X, color( E_TERM, true )+Z )
Multiply both sides by the left denominator.
\dfrac{ plus( A+X, B+Y ) }{ color( C, true ) } = \dfrac{ plus( D+X, E+Z ) }{ F }
color( C, true ) \cdot \dfrac{ plus( A+X, B+Y ) }{ color( C, true ) } = color( C, true ) \cdot \dfrac{ plus( D+X, E+Z ) }{ F }
plus( A+X, B+Y ) = color( C, true ) \cdot \dfrac { plus( D+X, E+Z ) }{ F }
Reduce the right side.
plus( A+X, B+Y ) = color( C, false ) \cdot \dfrac{ plus( D+X, E+Z ) }{ color( F, false ) }
plus( A+X, B+Y ) = color( C / F, false ) \cdot \left( plus( D+X, E+Z ) \right)
Multiply both sides by the right denominator.
plus( A+X, B+Y ) = C \cdot \dfrac{ plus( D+X, E+Z ) }{ color( F, false ) }
color( F, false ) \cdot \left( plus( A+X, B+Y ) \right) = color( F, false ) \cdot C \cdot \dfrac{ plus( D+X, E+Z ) }{ color( F, false ) }
color( F, false ) \cdot \left( plus( A+X, B+Y ) \right) = C \cdot \left( plus( D+X, E+Z ) \right)
Distribute the right sideboth sides.
plus( A+X, B+Y ) = color( C / F, true ) \cdot \left( plus( color( D+X, true ), color( E+Z, true ) ) \right)
color( F, true ) \cdot \left( plus( A+X, B+Y ) \right) = color( C, true ) \cdot \left( plus( D+X, E+Z ) \right)
plus( A_TERM+X, B_TERM+Y ) = plus( color( D_TERM, true )+X, color( E_TERM, true )+Z )
plus( color( A_TERM, true )+X, color( B_TERM, true )+Y ) = plus( color( D_TERM, true )+X, color( E_TERM, true )+Z )
Combine X
terms on the left.
plus( color( A_TERM+X, false ), B_TERM+Y ) = plus( color( D_TERM+X, false ), E_TERM+Z )
plus( color( (A_TERM-D_TERM)+X, false ), (B_TERM)+Y ) = (E_TERM)+Z
Move the Y
term to the right.
plus( (A_TERM-D_TERM)+X, color( B_TERM+Y, true ) ) = E_TERM+Z
(A_TERM-D_TERM)+X = plus( E_TERM+Z, color( (-B_TERM)+Y, true ) )
Isolate X
by dividing both sides by its coefficient.
color( A_TERM-D_TERM, false )+X = plus( E_TERM+Z, (-B_TERM)+Y )
X = \dfrac{ plus( E_TERM+Z, (-B_TERM)+Y ) }{ color( A_TERM-D_TERM, false ) }
All of these terms are divisible by GCD
.
Divide by the common factor and swap signs so the denominator isn't negative.
Divide by the common factor.
Swap signs so the denominator isn't negative.
X = \dfrac{ plus( color( round( E_TERM*DIVISOR ), true )+Z, color( round( -B_TERM*DIVISOR ), true )+Y ) }{ color( round( (A_TERM-D_TERM)*DIVISOR ), true ) }