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 x when its corresponding coefficients are negative inverses.
We can eliminate "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 ).