IB Maths AI HLNumber ToolkitPaper 2GDC solver~6 min read
Solving Equations using a GDC
Your GDC can solve equations that would be slow or impossible by hand. This note covers the two it solves directly — systems of linear equations and polynomial equations — how to enter them, and how to read and interpret the solutions it returns.
📘 What you need to know
A system of linear equations is n equations in n unknowns — no powers, no cross terms like xy.
The GDC’s algebra menu solves 2×2 and 3×3 systems — enter the equations and read off x, y (and z).
The point of intersection of two lines is the solution of the system formed by their two equations.
A polynomial equation has the form anxn + … + a1x + a0 = 0; the highest power n is its order.
A polynomial of order n has up to n real solutions — odd order gives at least one, even order may give none.
Graphing mode counts solutions: each point where the graph crosses the x-axis is one solution.
Systems of linear equations
A system of linear equations is a set of equations solved together. “Linear” means every term is a constant or a single variable to the power 1 — no squares, no products of variables. The GDC solves an n×n system: n equations in n unknowns.
Standard form of a linear system
2×2: ax + by = e , cx + dy = f3×3: ax + by + cz = p , … (three equations)rearrange each equation into this form before entering it into the GDC
To find the point of intersection of two straight lines, rearrange both into ax + by = e form, solve the 2×2 system, and write the answer as coordinates (x, y).
Polynomial equations
A polynomial equation is built from terms axk with k a non-negative integer, set equal to zero. The highest power is the order (or degree). Negative or fractional powers are not allowed.
Polynomial equation of order nanxn + an−1xn−1 + … + a1x + a0 = 0
n is a positive integer — e.g. 2x3 + 4x2 − x + 1 = 0 has order 3
The GDC’s polynomial solver asks for the order first, then the coefficients. It returns the roots — the values of x that satisfy the equation.
How many solutions?
A polynomial of order n has up to n real solutions. If n is odd there is always at least one; if n is even there may be none. To count them, plot y = … and see how often the graph meets the x-axis.
The cubic y = x³ − 7x + 6 meets the x-axis three times, so the equation x³ − 7x + 6 = 0 has three real solutions: x = −3, 1, 2.
Exam phrasing: if a question says “using technology, solve…”, you may go straight to the GDC — no algebraic working is needed, but state the equations clearly first.
🧠Recipe — solving an equation on the GDC
Identify the type: a system of linear equations, or a single polynomial equation.
Rearrange — linear equations into ax + by = e form; a polynomial into “… = 0”.
Open the solver: choose the system size n×n, or enter the polynomial’s order.
Enter the coefficients exactly as written, then read off the solutions the GDC gives.
Interpret — write intersections as coordinates, and reject any solution that doesn’t fit the context.
Worked examples
WE 1
A 2×2 linear system
Use technology to solve the system 4x + 3y = 27 and 2x − y = 1.
both equations are already in ax + by = e formenter into the GDC’s 2×2 solverread off the solutionx = 3, y = 5check: 4(3) + 3(5) = 27 and 2(3) − 5 = 1. ✓
WE 2
A 3×3 linear system
Solve the system x + y + z = 6, 2x − y + z = 3, x + 2y − z = 2.
3 equations, 3 unknowns — a 3×3 systementer all three into the GDC’s 3×3 solverread off the solutionx = 1, y = 2, z = 3enter the coefficients in the same order in every row, including any zeros.
WE 3
Point of intersection of two lines
Find the coordinates of the point where the lines y = 2x + 1 and y = −x + 7 intersect.
rearrange both into ax + by = e2x − y = −1 and x + y = 7solve the 2×2 system on the GDCx = 2, y = 5intersection at (2, 5)the solution of the system is the point of intersection — write it as coordinates.
WE 4
Solving a polynomial equation
Use your GDC to solve the equation x3 − 7x + 6 = 0.
a polynomial equation of order 3enter order 3 and the coefficients 1, 0, −7, 6the GDC returns the rootsx = −3, x = 1, x = 2don’t forget the coefficient of x2 is 0 — it still must be entered.
WE 5
How many solutions?
By considering the graph of y = x2 − 4x + 7, state how many real solutions the equation x2 − 4x + 7 = 0 has.
plot the graph on the GDCthe parabola has a minimum value of 3it stays entirely above the x-axisno crossings ⇒ no real solutions0 real solutionsorder 2 is even — so having no real solutions is possible.
WE 6
Full question: forming a linear system
At a café, 4 teas and 3 muffins cost $19, while 2 teas and 5 muffins cost $20. (a) Set up a system of linear equations. (b) Using technology, find the price of one tea and one muffin.
(a) let t = tea price, m = muffin price4t + 3m = 192t + 5m = 20(b) solve the 2×2 system on the GDCt = 2.5, m = 3(a) above · (b) tea $2.50, muffin $3.00define your variables clearly, then convert each sentence into one equation.
💡 Top tips
Rearrange first — linear equations into ax + by = e, polynomials into “= 0”.
Enter every coefficient, including any zeros for missing terms.
A point of intersection is just a 2×2 system — give the answer as coordinates.
Use graphing mode to count solutions before solving — one crossing per solution.
If a question says “using technology”, no algebraic working is required — but write the equations down.
âš Common mistakes
Not rearranging — entering y = 2x + 1 instead of 2x − y = −1.
Skipping a zero coefficient — x3 − 7x + 6 needs the 0 for the x2 term.
Choosing the wrong order for a polynomial — it is the highest power.
Leaving an intersection as just x — an intersection is a coordinate pair (x, y).
Assuming a polynomial always has n solutions — it has up to n; an even order may have none.
That completes the Number Toolkit. Next chapter: Sequences & Series — arithmetic and geometric patterns and their sums. The GDC stays close at hand: it solves the equations these problems often lead to, and lets you check answers quickly.
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