Solve for x and y using elimination.
\begin{align*}expr(["+", ["*", A1, "x"], ["*", B1, "y"]]) &= C1 \\
expr(["+", ["*", A2, "x"], ["*", B2, "y"]]) &= C2\end{align*}
x = X_NUMER / X_DENOM
y = Y_NUMER / Y_DENOM
We can eliminate ( XY_FLAG ? "x" : "y" ) when its corresponding coefficients are negative inverses.
Recalling our knowledge of least common multiples, multiply the top equation by MULT_1 and the bottom equation by MULT_2.
\begin{align*}expr(["+", ["*", A1*MULT_1, "x"], ["*", B1*MULT_1, "y"]]) &= C1*MULT_1\\
expr(["+", ["*", A2*MULT_2, "x"], ["*", B2*MULT_2, "y"]]) &= C2*MULT_2\end{align*}
Add the top and bottom equations.
expr(["*", B1*MULT_1+B2*MULT_2, "y"]) = C1*MULT_1+C2*MULT_2
Divide both sides by B1*MULT_1+B2*MULT_2 and reduce as necessary.
y = fractionReduce( Y_NUMER, Y_DENOM )
Substitute fractionReduce( Y_NUMER, Y_DENOM ) for y in the top equation.
expr(["+", ["*", A1, "x"], ["*", B1, " " + fractionReduce( Y_NUMER, Y_DENOM )]]) = C1
expr(["+", ["*", A1, "x"], fractionReduce( B1 * Y_NUMER, Y_DENOM )]) = C1
expr(["*", A1, "x"]) = fractionReduce( C1 * Y_DENOM - B1 * Y_NUMER, Y_DENOM )
x = fractionReduce(X_NUMER,X_DENOM)
The solution is\enspace x = fractionReduce( X_NUMER, X_DENOM ), \enspace y = fractionReduce( Y_NUMER, Y_DENOM ).
Recalling our knowledge of least common multiples, multiply the top equation by MULT_3 and the bottom equation by MULT_4.
\begin{align*}expr(["+", ["*", A1*MULT_3, "x"], ["*", B1*MULT_3, "y"]]) &= C1*MULT_3\\
expr(["+", ["*", A2*MULT_4, "x"], ["*", B2*MULT_4, "y"]]) &= C2*MULT_4\end{align*}
Add the top and bottom equations.
expr(["*", A1*MULT_3+A2*MULT_4, "x"]) = C1*MULT_3+C2*MULT_4
Divide both sides by A1*MULT_3+A2*MULT_4 and reduce as necessary.
x = fractionReduce( X_NUMER, X_DENOM )
Substitute fractionReduce( X_NUMER, X_DENOM ) for x in the top equation.
expr(["+", ["*", A1, " " + fractionReduce( X_NUMER, X_DENOM )], ["*", B1, "y"]]) = C1
expr(["+", fractionReduce( A1 * X_NUMER, X_DENOM ), ["*", B1, "y"]]) = C1
expr(["*", B1, "y"]) = fractionReduce( C1 * X_DENOM - A1 * X_NUMER, X_DENOM )
y = fractionReduce( Y_NUMER, Y_DENOM )
The solution is\enspace x = fractionReduce( X_NUMER, X_DENOM ), \enspace y = fractionReduce( Y_NUMER, Y_DENOM ).
Solve for x and y by deriving an expression for y from the second equation, and substituting it back into the first equation.
\begin{align*}expr(["+", ["*", A1, "x"], ["*", B1, "y"]]) &= C1 \\
expr(["+", ["*", A2, "x"], ["*", B2, "y"]]) &= C2\end{align*}
x = X_NUMER / X_DENOM
y = Y_NUMER / Y_DENOM
Begin by moving the x-term in the second equation to the right side of the equation.
expr(["*", B2, "y"]) = \color{BLUE}{expr(["+", ["*", -A2, "x"], C2])}
Divide both sides by B2 to isolate y.
y = \color{BLUE}{SIGN_1decimalFraction( -A2 / B2, "true", "true" )x + decimalFraction( C2 / B2, "true", "true" )}
Substitute this expression for y in the first equation.
expr(["*", A1, "x"])B1_SIGNabs( B1 )(\color{BLUE}{SIGN_1decimalFraction( -A2 / B2, "true", "true" )x + decimalFraction( C2 / B2, "true", "true" )}) = C1
expr(["*", A1, "x"]) + SIGN_2decimalFraction( -A2 / B2 * B1, "true", "true" )x + decimalFraction( C2 / B2 * B1, "true", "true" ) = C1
Simplify by combining terms, then solve for x.
decimalFraction( A1 + ( -A2 / B2 * B1 ), "true", "true" )x + decimalFraction( C2 / B2 * B1, "true", "true" ) = C1
decimalFraction( A1 + ( -A2 / B2 * B1 ), "true", "true" )x = decimalFraction( C1 - ( C2 / B2 * B1 ), "true", "true" )
x = fractionReduce( X_NUMER, X_DENOM )
Substitute fractionReduce( X_NUMER, X_DENOM ) for x back into the top equation.
expr(["+", ["*", A1, " " + fractionReduce( X_NUMER, X_DENOM )], ["*", B1, "y"]]) = C1
expr(["+", fractionReduce( A1 * X_NUMER, X_DENOM ), ["*", B1, "y"]]) = C1
expr(["*", B1, "y"]) = fractionReduce( C1 * X_DENOM - A1 * X_NUMER, X_DENOM )
y = fractionReduce( Y_NUMER, Y_DENOM )
The solution is\enspace x = fractionReduce( X_NUMER, X_DENOM ), \enspace y = fractionReduce( Y_NUMER, Y_DENOM ).
Solve for x and y by deriving an expression for x from the second equation, and substituting it back into the first equation.
\begin{align*}expr(["+", ["*", A1, "x"], ["*", B1, "y"]]) &= C1 \\
expr(["+", ["*", A2, "x"], ["*", B2, "y"]]) &= C2\end{align*}
x = X_NUMER / X_DENOM
y = Y_NUMER / Y_DENOM
Begin by moving the y-term in the second equation to the right side of the equation.
expr(["*", A2, "x"]) = \color{ORANGE}{expr(["+", ["*", -B2, "y"], C2])}
Divide both sides by A2 to isolate x.
x = \color{ORANGE}{SIGN_1decimalFraction( -B2 / A2, "true", "true" )y + decimalFraction( C2 / A2, "true", "true" )}
Substitute this expression for x in the first equation.
A1_SIGNabs( A1 )(\color{ORANGE}{SIGN_1decimalFraction( -B2 / A2, "true", "true" )y + decimalFraction( C2 / A2, "true", "true" )}) + expr(["*", B1, "y"]) = C1
SIGN_2decimalFraction( -B2 / A2 * A1, "true", "true" )y + decimalFraction( C2 / A2 * A1, "true", "true" ) + expr(["*", B1, "y"]) = C1
Simplify by combining terms, then solve for y.
decimalFraction( B1 + ( -B2 / A2 * A1 ), "true", "true" )y + decimalFraction( C2 / A2 * A1, "true", "true" ) = C1
decimalFraction( B1 + ( -B2 / A2 * A1 ), "true", "true" )y = decimalFraction( C1 - ( C2 / A2 * A1 ), "true", "true" )
y = fractionReduce( Y_NUMER, Y_DENOM )
Substitute fractionReduce( Y_NUMER, Y_DENOM ) for y in the top equation.
expr(["+", ["*", A1, "x"], ["*", B1, " " + fractionReduce( Y_NUMER, Y_DENOM )]]) = C1
expr(["+", ["*", A1, "x"], fractionReduce( B1 * Y_NUMER, Y_DENOM )]) = C1
expr(["*", A1, "x"]) = fractionReduce( C1 * Y_DENOM - B1 * Y_NUMER, Y_DENOM )
x = fractionReduce(X_NUMER,X_DENOM)
The solution is\enspace x = fractionReduce( X_NUMER, X_DENOM ), \enspace y = fractionReduce( Y_NUMER, Y_DENOM ).