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AI HL · 12.4

Solving Equations with Technology

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12.4.7

Mixed practice: equations and systems

12.4.7

Mixed practice: equations and systems

Target: I can form an equation or a pair of equations from a context and solve it with the calculator. (+4 more)

By the end of this section

Key points

Forming the equation

  • Say what the unknown is: "Let x be …". Use the units.
  • Write the fact from the question as an equation. Then move everything to one side to get a polynomial equal to zero.
  • Two unknowns need two equations. Three unknowns need three.
  • Common facts: area of a triangle ½bh; volume of a cuboid; surface area of a cube 6x²; handshakes n(n − 1)/2; consecutive integers m − 1, m, m + 1.

Solving with the calculator

  • Quadratics and cubics: use the polynomial solver. Systems: use the simultaneous equation solver.
  • Lengths, counts and times must be positive. Counts of people or objects must also be whole numbers. Reject roots that break this and say why.
  • Give 3 significant figures unless the question says otherwise. Do not round before the last step.

Formulae to use

  • Arithmetic: uₙ = u₁ + (n − 1)d and Sₙ = n/2 (2u₁ + (n − 1)d) = n/2 (u₁ + uₙ).
  • Geometric: u₁ + u₁r + u₁r² is the sum of three terms. Forming it gives a quadratic in r.
  • Cosine rule: a² = b² + c² − 2bc cos A. In an equation with x, expand and collect.
  • A system in cubes: let a = x³, b = y³, c = z³, solve for a, b, c, then take cube roots.

On the calculator

Solving a system of linear equations

  • Choose the system of equations tool with 2 or 3 equations.
  • Enter each equation with the unknowns on the left and the number on the right.
  • Read the solution for each unknown.

Screenshot and video to come

The polynomial solver

  • Write the equation as a polynomial equal to zero.
  • Enter the degree and the coefficients, including any zeros for missing powers.
  • Read every real root. Check each one against the question.

Screenshot and video to come

Worked examples

Problems like the ones in class. Pick one, try it, then open the working.
Three equations in cubes

Solve the simultaneous equations \(x^3 + y^3 + z^3 = −13,\ x^3 + 3y^3 + 2z^3 = −59,\ 2x^3 + y^3 − z^3 = −23\). Give \(x\) to 3 significant figures.

Working
  1. Substitute

    Let a = x³, b = y³, c = z³

    The equations are now linear in a, b and c.

  2. The system

    a + b + c = −13, a + 3b + 2c = −59, 2a + b − c = −23

    Enter it in the simultaneous equation solver as three equations in three unknowns.

  3. Solve for a, b, c

    From the calculator: a = 6, b = −27, c = 8

    This is the only step that needs the calculator.

  4. Cube roots

    x = ∛6 = 1.82, y = ∛(−27) = −3.00, z = ∛8 = 2.00

    A cube root of a negative number is negative.

Video walkthrough coming
Prices from two purchases

Maya buys 5 notebooks and 3 pens for €21.30. Ben buys 2 notebooks and 7 pens for €26.50. All notebooks cost the same and all pens cost the same. Find the price of one pen.

Working
  1. Define

    n = price of a notebook (€), p = price of a pen (€)

    Name the two unknowns first.

  2. Two equations

    5n + 3p = 21.3, 2n + 7p = 26.5

    One equation for each person's purchase.

  3. Solve

    From the calculator: n = 2.4, p = 3.1

    Simultaneous equation solver, two equations.

  4. Answer

    A pen costs €3.10

    Check: 5 × 2.40 + 3 × 3.10 = 21.30.

Video walkthrough coming
A fraction equation that becomes a cubic

Show that \(x^2 − 3 = \dfrac{5}{x + 1}\) can be written as \(x^{3} + x^{2} - 3 x - 8 = 0\). Hence solve the equation. Give the real solution to 3 significant figures.

Working
  1. Clear the fraction

    (x² − 3)(x + 1) = 5

    Multiply both sides by (x + 1).

  2. Expand

    x^{3} + x^{2} - 3 x - 3 = 5

    Expand every bracket.

  3. Zero on one side

    x^{3} + x^{2} - 3 x - 8 = 0

    This is the cubic.

  4. Solve

    Polynomial solver: x = 2.14

    The cubic has one real root. The other two roots are not real.

  5. Check

    x ≠ −1, so the root is allowed

    The original equation is not defined at x = −1.

Video walkthrough coming
Volume of a cuboid

A cuboid has edges \(x\) cm, \((x + 4)\) cm and \((3x − 2)\) cm. Its volume is 60 cm³. Find \(x\) to 3 significant figures.

x cm(x + 4) cm(3x − 2) cmABCDEFGH
Working
  1. Volume

    V = x(x + 4)(3x − 2) = 3 x^{3} + 10 x^{2} - 8 x

    Multiply the three edges, then expand.

  2. Equation

    3 x^{3} + 10 x^{2} - 8 x = 60, so 3 x^{3} + 10 x^{2} - 8 x - 60 = 0

    Put the volume on the right, then move it across.

  3. Solve

    Polynomial solver: x = 2.17

    The other roots are negative or not real, so they cannot be lengths.

  4. Check

    Edges: 2.17, 6.17, 4.50

    All three edges are positive.

Video walkthrough coming
First term and difference from two facts

An arithmetic sequence has \(S_{20} = 720\) and \(u_{20} = 74\). Find the common difference \(d\).

Working
  1. The sum

    S20 = 20/2 (u₁ + 74) = 720

    Use the sum formula with the first and last terms.

  2. Find u₁

    u₁ + 74 = 72, so u₁ = −2

    One unknown found.

  3. The last term

    u₁ + 19d = 74, so −2 + 19d = 74

    Use u_n = u₁ + (n − 1)d.

  4. Solve

    19d = 76, so d = 4

    Or enter both equations in the simultaneous equation solver.

Video walkthrough coming