Solving linear equations with distribution and combining like terms. The coefficient to be distributed is chosen thusly: - distribute positive numbers 22.5% of the time - distribute a minus sign 40% of the time - distribute other negative numbers 37.5% of the time Problem types: prob1 A = Bx + C(Dx + E) prob2 A = B(C + Dx) + Ex prob3 A(Bx + C) = D(E − Fx) + Gx prob4 Ax + B(Cx + D) = Ex + F prob5 A + Bx = C(Dx + E) prob6 Ax + B(C + Dx) = E prob7 A(Bx + C) = Dx + E prob8 A(B + Cx) + D = E + Fx prob9 A(Bx + C) = D(E + Fx)
randFromArray("abkmnpvx") randRangeNonZero(-10, 10) randRangeNonZero(-10, 10) randRangeWeightedExclude(-6, 4, -1, 0.4, [0, 1]) randRangeNonZero(-10, 10) randRange(1, 3) (A - B * C) / (B + C * D)

Solve for X:

\qquad A = expr(["+", ["*", B, X], ["*", C, ["+", ["*", D, X], E]]])

X = SOLUTION

Try simplifying the right side of the equation before solving it.

Distribute the negative in front of the parentheses. Be careful! The negative sign in front of the parentheses means we're multiplying by \pink{-1}:

Distribute the \pink{C}. Be careful to pay attention to the negative sign when you distribute:

Distribute the \pink{C}:

\qquad\begin{eqnarray} A &=& expr(["*", B, X]) + \pink{C}\blue{(expr(["+", ["*", D, X], E]))} \\ \\ A &=& expr(["*", B, X]) + \pink{(C)}\blue{(expr(["*", D, X]))} + \pink{(C)}\blue{(E)} \end{eqnarray}

Multiply:

\qquad A = expr(["+", ["*", B, X], ["*", C * D, X], C * E])

Combine the X terms:

\qquad\begin{eqnarray} A &=& \blue{expr(["+", ["*", B, X], ["*", C * D, X]])} + C * E \\ \\ A &=& \blue{expr(["*", B + C * D, X])} + B * C \end{eqnarray}

B * C < 0 ? "Add" : "Subtract" \green{abs(B * C)} B * C < 0 ? "to" : "from" both sides to isolate the X term on the right side:

\qquad\begin{eqnarray} A &=& expr(["*", B + C * D, X]) \green{{} + B * C} \\ \\ \green{{}+-B * C} && \green{{}+-B * C} \\ \\ A - B * C &=& expr(["*", B + C * D, X]) \end{eqnarray}

Divide both sides by \green{B + C * D} to isolate X:

\qquad\begin{eqnarray} A - B * C &=& expr(["*", B + C * D, X]) \\ \\ \dfrac{A - B * C} {\green{B + C * D}} &=& \dfrac{\green{\cancel{B + C * D}} X}{\green{\cancel{B + C * D}}} \end{eqnarray}

Simplify:

\qquad fractionReduce(A - B * C, B + C * D) = X

randFromArray("abkmnpvx") randRangeNonZero(-10, 10) randRangeWeightedExclude(-6, 4, -1, 0.4, [0, 1]) randRange(1, 3) randRangeNonZero(-10, 10) randRangeNonZero(-10, 10) (A - B * C) / (B * D + E)

Solve for X:

\qquad A = expr(["+", ["*", B, ["+", C, ["*", D, X]]], ["*", E, X]])

X = SOLUTION

Try simplifying the right side of the equation before solving it.

Distribute the negative in front of the parentheses. Be careful! The negative sign in front of the parentheses means we're multiplying by \pink{-1}:

Distribute the \pink{B}. Be careful to pay attention to the negative sign when you distribute:

Distribute the \pink{B}:

\qquad\begin{eqnarray} A &=& \pink{B}\blue{(expr(["+", C, ["*", D, X]]))} + expr(["*", E, X]) \\ \\ A &=& \pink{(B)}\blue{(C)} + \pink{(B)}\blue{(expr(["*", D, X]))} + expr(["*", E, X]) \end{eqnarray}

Multiply:

\qquad A = expr(["+", B * C, ["*", B * D, X], ["*", E, X]])

Combine the X terms:

\qquad\begin{eqnarray} A &=& B * C \blue{{} + expr(["*", B * D, X])} \blue{{} + expr(["*", E, X])} \\ \\ A &=& B * C \blue{{} + expr(["*", B * D + E, X])} \end{eqnarray}

B * C < 0 ? "Add" : "Subtract" \green{abs(B * C)} B * C < 0 ? "to" : "from" both sides to isolate the X term on the right side:

\qquad\begin{eqnarray} A &=& \green{B * C} + expr(["*", B * D + E, X]) \\ \\ \green{{}+-B * C} && \green{{}+-B * C} \\ \\ A - B * C &=& expr(["*", B * D + E, X]) \end{eqnarray}

Divide both sides by \green{B * D + E} to isolate X:

\qquad\begin{eqnarray} A - B * C &=& expr(["*", B * D + E, X]) \\ \\ \dfrac{A - B * C} {\green{B * D + E}} &=& \dfrac{\green{\cancel{B * D + E}} X}{\green{\cancel{B * D + E}}} \end{eqnarray}

Simplify:

\qquad fractionReduce(A - B * C, B * D + E) = X

randFromArray("abkmnpvx") randRangeWeightedExclude(-6, 4, -1, 0.4, [0, 1]) randRangeNonZero(-10, 10) randRange(1, 3) randRangeWeightedExclude(-6, 4, -1, 0.4, [0, 1]) randRangeNonZero(-10, 10) randRangeNonZero(-10, 10) randRangeNonZero(-10, 10) (D * E - A * C) / ((A * B) - (D * F + G))

Solve for X:

\qquad expr(["*", A, ["+", ["*", B, X], C]]) = expr(["+", ["*", D, ["+", E, ["*", F, X]]], ["*", G, X]])

X = SOLUTION

Try simplifying each side of the equation before solving it.

Distribute the negative in front of the parentheses. Be careful! The negative sign in front of the parentheses means we're multiplying by \pink{-1}:

Distribute the \pink{A}. Be careful to pay attention to the negative sign when you distribute:

Distribute the \pink{A}:

\qquad\begin{eqnarray} \pink{A}\blue{(expr(["+", ["*", B, X], C]))} &=& expr(["+", ["*", D, ["+", E, ["*", F, X]]], ["*", G, X]]) \\ \\ \pink{(A)}\blue{(expr(["*", B, X]))} + \pink{(A)}\blue{(C)} &=& expr(["+", ["*", D, ["+", E, ["*", F, X]]], ["*", G, X]]) \end{eqnarray}

Multiply:

\qquad expr(["+", ["*", A * B, X], A * C]) = expr(["+", ["*", D, ["+", E, ["*", F, X]]], ["*", G, X]])

Distribute the negative in front of the parentheses. Be careful! The negative sign in front of the parentheses means we're multiplying by \pink{-1}:

Distribute the \pink{D}. Be careful to pay attention to the negative sign when you distribute:

Distribute the \pink{D}:

\qquad\begin{eqnarray} expr(["+", ["*", A * B, X], A * C]) &=& \pink{(D)}\blue{(expr(["+", E, ["*", F, X]]))} + expr(["*", G, X]) \\ \\ expr(["+", ["*", A * B, X], A * C]) &=& \pink{(D)} \blue{(E)} + \pink{(D)} \blue{(expr(["*", F, X]))} + expr(["*", G, X]) \end{eqnarray}

Multiply:

\qquad expr(["+", ["*", A * B, X], A * C]) = expr(["+", D * E, ["*", D * F, X], ["*", G, X]])

Combine the X terms:

\qquad\begin{eqnarray} expr(["+", ["*", A * B, X], A * C]) &=& D * E \blue{{} + expr(["+", ["*", D * F, X], ["*", G, X]])} \\ \\ expr(["+", ["*", A * B, X], A * C]) &=& D * E \blue{{} + expr(["*", D * F + G, X])} \end{eqnarray}

D * F + G < 0 ? "Add" : "Subtract" \green{expr(["*", abs(D * F + G), X])} D * F + G < 0 ? "to" : "from" both sides to eliminate the X term from the right side:

\qquad\begin{eqnarray} expr(["+", ["*", A * B, X], A * C]) &=& D * E \green{{} + expr(["*", D * F + G, X])} \\ \\ \green{{}+expr(["*", -(D * F + G), X])} && \green{{} + expr(["*", -(D * F + G), X])} \\ \\ expr(["+", ["*", A * B, X], A * C]) \green{+expr(["*", -(D * F + G), X])} &=& D * E \end{eqnarray}

Combine the X terms:

\qquad\begin{eqnarray} \blue{expr(["*", A * B, X])} + A * C \blue{+expr(["*", -(D * F + G), X])} &=& D * E \\ \\ \blue{expr(["*", (A * B) - (D * F + G), X])} + A * C &=& D * E \end{eqnarray}

A * C < 0 ? "Add" : "Subtract" \green{abs(A * C)} A * C < 0 ? "to" : "from" both sides to isolate the X term on the left side:

\qquad\begin{eqnarray} expr(["*", (A * B) - (D * F + G), X]) + \green{A * C} &=& D * E \\ \\ \green{{}+-A * C} && \green{{}+-A * C} \\ \\ expr(["*", (A * B) - (D * F + G), X]) &=& D * E - A * C \end{eqnarray}

Divide both sides by \green{(A * B) - (D * F + G)} to isolate X:

\qquad\begin{eqnarray} expr(["*", (A * B) - (D * F + G), X]) &=& D * E - A * C \\ \\ \dfrac{\green{\cancel{(A * B) - (D * F + G)}} X}{\green{\cancel{(A * B) - (D * F + G)}}} &=& \dfrac{D * E - A * C} {\green{(A * B) - (D * F + G)}} \\ \\ \end{eqnarray}

Simplify:

\qquad X = fractionReduce(D * E - A * C, (A * B) - (D * F + G))

randFromArray("abkmnpvx") randRangeNonZero(-8, 8) randRangeWeightedExclude(-6, 4, -1, 0.4, [0, 1]) randRange(1, 3) randRangeNonZero(-10, 10) randRangeNonZero(-6, 6) randRangeNonZero(-10, 10) (F - B * D) / (A + B * C - E)

Solve for X:

\qquad expr(["+", ["*", A, X], ["*", B, ["+", ["*", C, X], D]]]) = expr(["+", ["*", E, X], F])

X = SOLUTION

Try simplifying the left side of the equation before solving it.

Distribute the negative in front of the parentheses. Be careful! The negative sign in front of the parentheses means we're multiplying by \pink{-1}:

Distribute the \pink{B}. Be careful to pay attention to the negative sign when you distribute:

Distribute the \pink{B}:

\qquad\begin{eqnarray} expr(["*", A, X]) + \pink{B} \blue{(expr(["+", ["*", C, X], D]))} &=& expr(["+", ["*", E, X], F]) \\ \\ expr(["*", A, X]) + \pink{(B)} \blue{(expr(["*", C, X]))} + \pink{(B)} \blue{(D)} &=& expr(["+", ["*", E, X], F]) \end{eqnarray}

Multiply:

\qquad expr(["+", ["*", A, X], ["*", B * C, X], B * D]) = expr(["+", ["*", E, X], F])

E < 0 ? "Add" : "Subtract" \green{expr(["*", abs(E), X])} E < 0 ? "to" : "from" both sides to eliminate the X term from the right side:

\qquad\begin{eqnarray} expr(["+", ["*", A, X], ["*", B * C, X], B * D]) &=& \green{expr(["*", E, X])} + F \\ \\ \green{{}+expr(["*", -E, X])} &=& \green{{} + expr(["*", -E, X])} \\ \\ expr(["+", ["*", A, X], ["*", B * C, X], B * D]) \green{ +expr(["*", -E, X])} &=& F \end{eqnarray}

Combine the X terms:

\qquad\begin{eqnarray} \blue{expr(["+", ["*", A, X], ["*", B * C, X]])} + B * D \blue{+expr(["*", -E, X])} &=& F \\ \\ \blue{expr(["*", A + B * C - E, X])} + B * D & = & F \end{eqnarray}

B * D < 0 ? "Add" : "Subtract" \green{abs(B * D)} B * D < 0 ? "to" : "from" both sides to isolate the X term on the left side:

\qquad\begin{eqnarray} expr(["*", A + B * C - E, X]) \green{+ B * D} & = & F \\ \\ \green{{}+-B * D} && \green{{}+-B * D} \\ \\ expr(["*", A + B * C - E, X]) &=& F - B * D \end{eqnarray}

Divide both sides by \green{A + B * C - E} to isolate X:

\qquad\begin{eqnarray} \green{A + B * C - E}X &=& F - B * D \\ \\ \dfrac{\green{\cancel{A + B * C - E}} X}{\green{\cancel{A + B * C - E}}} &=& \dfrac{F - B * D} {\green{A + B * C - E}} \\ \\ \end{eqnarray}

Simplify:

\qquad X = fractionReduce(F - B * D, A + B * C - E)

randFromArray("abkmnpvx") randRangeNonZero(-8, 8) randRangeNonZero(-10, 10) randRangeWeightedExclude(-6, 4, -1, 0.4, [0, 1]) randRange(1, 3) randRangeNonZero(-6, 6) (C * E - A) / (B - C * D)

Solve for X:

\qquad expr(["+", A, ["*", B, X]]) = expr(["*", C, ["+", ["*", D, X], E]])

X = SOLUTION

Try simplifying the right side of the equation before solving it.

Distribute the negative in front of the parentheses. Be careful! The negative sign in front of the parentheses means we're multiplying by \pink{-1}:

Distribute the \pink{C}. Be careful to pay attention to the negative sign when you distribute:

Distribute the \pink{C}:

\qquad\begin{eqnarray} expr(["+", A, ["*", B, X]]) &=& \pink{C}\blue{(expr(["+", ["*", D, X], E]))} \\ \\ expr(["+", A, ["*", B, X]]) &=& \pink{(C)}\blue{(expr(["*", D, X]))} + \pink{(C)}\blue{(E)} \end{eqnarray}

Multiply:

\qquad expr(["+", A, ["*", B, X]]) = expr(["+", ["*", C * D, X], C * E])

C * D < 0 ? "Add" : "Subtract" \green{expr(["*", abs(C * D), X])} C * D < 0 ? "to" : "from" both sides to eliminate the X term from the right side:

\qquad\begin{eqnarray} expr(["+", A, ["*", B, X]]) &=& \green{expr(["*", C * D, X])} + C * E \\ \\ \green{{}+expr(["*", -C * D, X])} && \green{{} + expr(["*", -C * D, X])} \\ \\ expr(["+", A, ["*", B, X]]) \green{{} + expr(["*", -C * D, X])} &=& C * E \end{eqnarray}

Combine the X terms:

\qquad\begin{eqnarray} A \blue{{} + expr(["+", ["*", B, X], ["*", -C * D, X]])} &=& C * E \\ \\ A \blue{{} + expr(["*", B - C * D, X])} &=& C * E \end{eqnarray}

A < 0 ? "Add" : "Subtract" \green{abs(A)} A < 0 ? "to" : "from" both sides to isolate the X term on the left side:

\qquad\begin{eqnarray} \green{A} + expr(["*", B - C * D, X]) &=& C * E \\ \\ \green{{}+-A} && \green{{}+-A} \\ \\ expr(["*", B - C * D, X]) &=& C * E - A \end{eqnarray}

Divide both sides by \green{B - C * D} to isolate X:

\qquad\begin{eqnarray} \green{B - C * D}X &=& C * E - A \\ \\ \dfrac{\green{\cancel{B - C * D}} X}{\green{\cancel{B - C * D}}} &=& \dfrac{C * E - A} {\green{B - C * D}} \\ \\ \end{eqnarray}

Simplify:

\qquad X = fractionReduce(C * E - A, B - C * D)

randFromArray("abkmnpvx") randRangeNonZero(-8, 8) randRangeWeightedExclude(-6, 4, -1, 0.4, [0, 1]) randRangeNonZero(-10, 10) randRangeNonZero(-6, 6) randRangeNonZero(-10, 10) (E - B * C) / (A + B * D)

Solve for X:

\qquad expr(["+", ["*", A, X], ["*", B, ["+", C, ["*", D, X]]]]) = E

X = SOLUTION

Try simplifying the left side of the equation before solving it.

Distribute the negative in front of the parentheses. Be careful! The negative sign in front of the parentheses means we're multiplying by \pink{-1}:

Distribute the \pink{B}. Be careful to pay attention to the negative sign when you distribute:

Distribute the \pink{B}:

\qquad\begin{eqnarray} expr(["*", A, X]) + \pink{B}\blue{(expr(["+", C, ["*", D, X]]))} &=& E \\ \\ expr(["*", A, X]) + \pink{(B)}\blue{(C)} + \pink{(B)}\blue{(expr(["*", D, X]))} &=& E \end{eqnarray}

Multiply:

\qquad expr(["+", ["*", A, X], B * C, ["*", B * D, X]]) = E

Combine the X terms:

\qquad\begin{eqnarray} \blue{expr(["*", A, X])} + B * C \blue{{}+ expr(["*", B * D, X])} &=& E \\ \\ \blue{expr(["*", A + B * D, X])} + B * C &=& E \end{eqnarray}

B * C < 0 ? "Add" : "Subtract" \green{abs(B * C)} B * C < 0 ? "to" : "from" both sides to isolate the X term on the left side:

\qquad\begin{eqnarray} expr(["*", A + B * D, X]) \green{{} + B * C} &=& E \\ \\ \green{{}+-B * C} && \green{{}+-B * C} \\ \\ expr(["*", A + B * D, X]) &=& E - B * C \end{eqnarray}

Divide both sides by \green{A + B * D} to isolate X:

\qquad\begin{eqnarray} \green{A + B * D}X &=& E - B * C \\ \\ \dfrac{\green{\cancel{A + B * D}} X}{\green{\cancel{A + B * D}}} &=& \dfrac{E - B * C} {\green{A + B * D}} \\ \\ \end{eqnarray}

Simplify:

\qquad X = fractionReduce(E - B * C, A + B * D)

randFromArray("abkmnpvx") randRangeWeightedExclude(-6, 4, -1, 0.4, [0, 1]) randRangeNonZero(-10, 10) randRange(1, 3) randRangeNonZero(-10, 10) randRangeNonZero(-10, 10) (E - A * C) / (A * B - D)

Solve for X:

\qquad expr(["*", A, ["+", ["*", B, X], C]]) = expr(["+", ["*", D, X], E])

X = SOLUTION

Try simplifying the left side of the equation before solving it.

Distribute the negative in front of the parentheses. Be careful! The negative sign in front of the parentheses means we're multiplying by \pink{-1}:

Distribute the \pink{A}. Be careful to pay attention to the negative sign when you distribute:

Distribute the \pink{A}:

\qquad\begin{eqnarray} \pink{A}\blue{(expr(["+", ["*", B, X], C]))} &=& expr(["+", ["*", D, X], E]) \\ \\ \pink{(A)}\blue{(expr(["*", B, X]))} + \pink{(A)}\blue{(C)} &=& expr(["+", ["*", D, X], E]) \end{eqnarray}

Multiply:

\qquad expr(["+", ["*", A * B, X], A * C]) = expr(["+", ["*", D, X], E])

D < 0 ? "Add" : "Subtract" \green{expr(["*", abs(D), X])} D < 0 ? "to" : "from" both sides to eliminate the X term from the right side:

\qquad\begin{eqnarray} expr(["+", ["*", A * B, X], A * C]) &=& \green{expr(["*", D, X])} + E \\ \\ \green{{}+expr(["*", -D, X])} && \green{{} + expr(["*", -D, X])} \\ \\ expr(["+", ["*", A * B, X], A * C]) \green{{} + expr(["*", -D, X])} &=& E \end{eqnarray}

Combine the X terms:

\qquad\begin{eqnarray} \blue{expr(["*", A * B, X])} + A * C \blue{{}+ expr(["*", -D, X])} &=& E \\ \\ \blue{expr(["*", A * B - D, X])} + A * C &=& E \end{eqnarray}

A * C < 0 ? "Add" : "Subtract" \green{abs(A * C)} A * C < 0 ? "to" : "from" both sides to isolate the X term on the left side:

\qquad\begin{eqnarray} expr(["*", A * B - D, X]) \green{{} + A * C} &=& E \\ \\ \green{{}+-A * C} && \green{{}+-A * C} \\ \\ expr(["*", A * B - D, X]) &=& E - A * C \end{eqnarray}

Divide both sides by \green{A * B - D} to isolate X:

\qquad\begin{eqnarray} \green{A * B - D}X &=& E - A * C \\ \\ \dfrac{\green{\cancel{A * B - D}} X}{\green{\cancel{A * B - D}}} &=& \dfrac{E - A * C} {\green{A * B - D}} \\ \\ \end{eqnarray}

Simplify:

\qquad X = fractionReduce(E - A * C, A * B - D)

randFromArray("abkmnpvx") randRangeWeightedExclude(-6, 4, -1, 0.4, [0, 1]) randRangeNonZero(-10, 10) randRange(1, 3) randRangeNonZero(-10, 10) randRangeNonZero(-10, 10) randRangeNonZero(-10, 10) (E - A * C - D) / (A * B - F)

Solve for X:

\qquad expr(["+", ["*", A, ["+", ["*", B, X], C]], D]) = expr(["+", E, ["*", F, X]])

X = SOLUTION

Try simplifying the left side of the equation before solving it.

Distribute the negative in front of the parentheses. Be careful! The negative sign in front of the parentheses means we're multiplying by \pink{-1}:

Distribute the \pink{A}. Be careful to pay attention to the negative sign when you distribute:

Distribute the \pink{A}:

\qquad\begin{eqnarray} \pink{A}\blue{(expr(["+", ["*", B, X], C]))} + D &=& expr(["+", E, ["*", F, X]]) \\ \\ \pink{(A)}\blue{(expr(["*", B, X]))} + \pink{(A)}\blue{(C)} + D &=& expr(["+", E, ["*", F, X]]) \end{eqnarray}

Multiply:

\qquad expr(["+", ["*", A * B, X], A * C, D]) = expr(["+", E, ["*", F, X]])

F < 0 ? "Add" : "Subtract" \green{expr(["*", abs(F), X])} F < 0 ? "to" : "from" both sides to eliminate the X term from the right side:

\qquad\begin{eqnarray} expr(["+", ["*", A * B, X], A * C, D]) &=& E \green{{} + expr(["*", F, X])} \\ \\ \green{{}+expr(["*", -F, X])} && \green{{} + expr(["*", -F, X])} \\ \\ expr(["+", ["*", A * B, X], A * C, D]) \green{{}+expr(["*", -F, X])} &=& E \end{eqnarray}

Combine the X terms:

\qquad\begin{eqnarray} \blue{expr(["*", A * B, X])} + expr(["+", A * C, D]) \blue{{} + expr(["*", -F, X])} &=& E \\ \\ expr(["+", A * C, D]) \blue{{} + expr(["*", A * B - F, X])} &=& E \end{eqnarray}

Combine the numeric terms:

\qquad\begin{eqnarray} \blue{expr(["+", A * C, D])} + expr(["*", A * B - F, X]) &=& E \\ \\ \blue{A * C + D} + expr(["*", A * B - F, X]) &=& E \end{eqnarray}

A * C + D < 0 ? "Add" : "Subtract" \green{abs(A * C + D)} A * C + D < 0 ? "to" : "from" both sides to isolate the X term on the left side:

\qquad\begin{eqnarray} \green{A * C + D} + expr(["*", A * B - F, X]) &=& E \\ \\ \green{{}+-(A * C + D)} && \green{{}+-(A * C + D)} \\ \\ expr(["*", A * B - F, X]) &=& E - A * C - D \end{eqnarray}

Divide both sides by \green{A * B - F} to isolate X:

\qquad\begin{eqnarray} \green{A * B - F}X &=& E - A * C - D \\ \\ \dfrac{\green{\cancel{A * B - F}} X}{\green{\cancel{A * B - F}}} &=& \dfrac{E - A * C - D} {\green{A * B - F}} \\ \\ \end{eqnarray}

Simplify:

\qquad X = fractionReduce(E - A * C - D, A * B - F)

randFromArray("abkmnpvx") randRangeWeightedExclude(-6, 4, -1, 0.4, [0, 1]) randRangeNonZero(-10, 10) randRange(1, 3) randRangeWeightedExclude(-6, 4, -1, 0.4, [0, 1]) randRangeNonZero(-10, 10) randRangeNonZero(-10, 10) (D * E - A * C) / (A * B - D * F)

Solve for X:

\qquad expr(["*", A, ["+", ["*", B, X], C]]) = expr(["*", D, ["+", E, ["*", F, X]]])

X = SOLUTION

Try simplifying each side of the equation before solving it.

Distribute the negative in front of the parentheses. Be careful! The negative sign in front of the parentheses means we're multiplying by \pink{-1}:

Distribute the \pink{A}. Be careful to pay attention to the negative sign when you distribute:

Distribute the \pink{A}:

\qquad\begin{eqnarray} \pink{A}\blue{(expr(["+", ["*", B, X], C]))} &=& expr(["*", D, ["+", E, ["*", F, X]]]) \\ \\ \pink{(A)}\blue{(expr(["*", B, X]))} + \pink{(A)}\blue{(C)} &=& expr(["*", D, ["+", E, ["*", F, X]]]) \end{eqnarray}

Multiply:

\qquad expr(["+", ["*", A * B, X], A * C]) = expr(["*", D, ["+", E, ["*", F, X]]])

Distribute the negative in front of the parentheses. Be careful! The negative sign in front of the parentheses means we're multiplying by \pink{-1}:

Distribute the \pink{D}. Be careful to pay attention to the negative sign when you distribute:

Distribute the \pink{D}:

\qquad\begin{eqnarray} expr(["+", ["*", A * B, X], A * C]) &=& \pink{(D)}\blue{(expr(["+", E, ["*", F, X]]))} \\ \\ expr(["+", ["*", A * B, X], A * C]) &=& \pink{(D)} \blue{(E)} + \pink{(D)} \blue{(expr(["*", F, X]))} \end{eqnarray}

Multiply:

\qquad expr(["+", ["*", A * B, X], A * C]) = expr(["+", D * E, ["*", D * F, X]])

D * F < 0 ? "Add" : "Subtract" \green{expr(["*", abs(D * F), X])} D * F < 0 ? "to" : "from" both sides to eliminate the X term from the right side:

\qquad\begin{eqnarray} expr(["+", ["*", A * B, X], A * C]) &=& D * E \green{{} + expr(["*", D * F, X])} \\ \\ \green{{}+expr(["*", -(D * F), X])} && \green{{} + expr(["*", -(D * F), X])} \\ \\ expr(["+", ["*", A * B, X], A * C]) \green{+expr(["*", -(D * F), X])} &=& D * E \end{eqnarray}

Combine the X terms:

\qquad\begin{eqnarray} \blue{expr(["*", A * B, X])} + A * C \blue{+expr(["*", -(D * F), X])} &=& D * E \\ \\ \blue{expr(["*", A * B - D * F, X])} + A * C &=& D * E \end{eqnarray}

A * C < 0 ? "Add" : "Subtract" \green{abs(A * C)} A * C < 0 ? "to" : "from" both sides to isolate the X term on the left side:

\qquad\begin{eqnarray} expr(["*", A * B - D * F, X]) + \green{A * C} &=& D * E \\ \\ \green{{}+-A * C} && \green{{}+-A * C} \\ \\ expr(["*", A * B - D * F, X]) &=& D * E - A * C \end{eqnarray}

Divide both sides by \green{A * B - D * F} to isolate X:

\qquad\begin{eqnarray} expr(["*", A * B - D * F, X]) &=& D * E - A * C \\ \\ \dfrac{\green{\cancel{A * B - D * F}} X}{\green{\cancel{A * B - D * F}}} &=& \dfrac{D * E - A * C} {\green{A * B - D * F}} \\ \\ \end{eqnarray}

Simplify:

\qquad X = fractionReduce(D * E - A * C, A * B - D * F)