Fundamental Mathematics I: send your chapter so we can check whether we can help before you book.
Concordia MATH 208, Fundamental Mathematics I, connects algebra with quantitative business models. Its Fall 2026 outline moves from functions and financial calculations to matrix methods, resource constraints, counting and probability. The example below develops a production decision from a system of inequalities.
Review fractions, percentages, exponents, equation solving and graphing a line. Practise naming a variable and its units before translating a sentence into mathematics. For financial exercises, distinguish the annual rate from the rate per payment or compounding period. These are study suggestions, not an eligibility assessment.
The current Concordia calendar lists MATH 206 or equivalent as the prerequisite. Credit and program restrictions also apply; Concordia advising determines equivalencies and eligibility. MATH 209 is linked here as a related course, not a stated prerequisite.
Check the official course catalogueLet and be two unknown real quantities satisfying and . Find both quantities, then determine whether the same pair also meets the limit .
1. Eliminate one unknown. Multiply the second equation by to obtain . Subtract it from the first equation: . The two terms containing cancel, leaving . 2. Recover the other unknown. Substitute into : , so . The solution pair is . 3. Verify the original equalities. They give and , so both hold. 4. Check the additional limit separately. For this pair, , and . Thus the pair meets the inequality, with a gap of below the limit. If the inequality represents a capacity constraint, that gap is one unit of unused capacity. Solving the equalities alone would not establish whether an additional limit is met.
A resource constraint is a limit, not a promise to use every available hour. This production example connects a small matrix calculation with the graphical method of linear programming.
A workshop chooses weekly quantities of standard batches and of premium batches. A standard batch needs assembly hours and finishing hour; a premium batch needs and , respectively. There are assembly hours and finishing hours available. Assume all produced batches can be sold and each contributes or dollars after variable costs. Let denote total weekly contribution, in dollars: sales revenue less variable production costs. Each standard batch contributes 30 dollars, so batches contribute ; premium batches similarly contribute . Thus contributions add.
The assembly hours used are : multiply each batch count by its hours per batch, then add. This total cannot exceed 120. Finishing similarly uses hours, at most 70. Counts cannot be negative. A feasible plan is a pair satisfying all four inequalities at once. The model is subject to Fixed costs do not affect which mix maximizes contribution if they are the same for every mix. We first allow fractional batch quantities; the optimum found below also has whole-number batches.
A boundary is where one inequality becomes an equality. A corner can occur where two boundaries meet, so first find the meeting point of the two resource lines. In the matrix below, the first and second columns are the coefficients of and ; the column after the bar contains the available hours. Subtracting a multiple of one equation from another preserves their simultaneous solutions. At the intersection where both resources are fully used, solve the boundary equalities. The augmented matrix is Replacing row 1 by row 1 minus twice row 2 gives Thus , then . Check: and . This finds one candidate; it does not yet prove that this candidate is best.
The nonnegative axes are also boundaries. On the horizontal axis set : the constraints give both and , so use the stricter limit 60. On the vertical axis set : both and must hold, so use 35. The feasible polygon therefore has corners , , and . The other intercepts, and , violate a constraint. Evaluate This polygon contains feasible points and is bounded: it does not extend indefinitely. A linear objective, such as the total contribution here, attains a maximum at one of its corners, so checking these corners is sufficient. Geometrically, a linear objective has no isolated maximum strictly inside this polygon: along any segment its value changes at a constant rate. Every point in the polygon is a weighted average of corners, so its objective value cannot exceed the largest corner value.
Choose standard batches and premium batches, for a maximum weekly contribution of dollars before fixed costs. Both resource limits are reached in this particular optimum. That conclusion comes from comparing all corners, not from a general rule that every resource must be exhausted. If indivisible batches produced a fractional optimum, simply rounding could break feasibility or miss the best integer choice.
Under the original constraints, test and ; compute the contribution only for feasible plans.
uses assembly hours but finishing hours, so it is infeasible. The plan uses and hours and contributes dollars.
Change the available hours to for assembly and for finishing, keeping . Find all feasible corners and the best mix.
The new boundaries are and . Subtract twice the second equation from the first: , hence and . On the axes, both constraints give and . The corners are , , , . Their contributions are , , and , respectively; choose .
Return to the original capacities but use . Find the optimum and any unused capacity.
Only the objective changed, so the feasible region and its four corners stay the same. Re-evaluate at those corners rather than solving the intersection again. At the original corners, the new contributions are , , and . Choose : assembly uses all hours, while finishing uses of , leaving hours unused. At the optimum, some of a resource can remain unused.
For the next resource-allocation problem, write the inequalities first. Use elimination to locate boundary intersections, reject infeasible points and compare the objective at every remaining corner.
Official Concordia MATH 208 outline, Fall 2026
Your tutor can work through your setup to connect matrix rows and columns to the quantities in the problem, then locate an error in elimination or in a resource constraint. For probability, you can compare possible sample spaces together and practise explaining why a counting or conditioning method fits.
Use your current section outline to choose the scope. Mix finance, systems, inequalities and probability so the method is not supplied by the exercise order. Rework one mistake without its solution, then change one condition and explain how the method or answer changes. Follow your section’s rules for calculators and permitted aids.
We work through permitted practice and comparable problems, without completing graded assignments or exams on your behalf.
Understand reversible row operations, solve and check a linear system, and distinguish one, no and infinitely many solutions. Original practice for Concordia MATH 204 and McGill MATH 133.
Ten essential algebra skills with worked examples, common mistakes and self-checks: a practical starting point before university or CEGEP calculus.
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