HSC · Mathematics Standard 1 · Algebra · also VCE Graphs & Relations live from the app

Linear relationships

Every non-vertical straight line has the rule y = mx + c: the gradient m is the rate of change (how much y changes for each step of 1 in x) and c is the y-intercept, the value of y when x = 0. Reading a graph runs the same way in reverse: c is where the line cuts the vertical axis and m is the rise for one unit of run. Put two lines on the same axes and the pair has at most one shared point. Finding it is solving simultaneous equations: set the two rules equal, solve for x, substitute back for y. This is the real knowscape from knowhere, not a picture of one. Drag it. Watch what actually changes.

mathematics standard 1 · algebra · linear relationshipsdrag it · it is yours
the one idea

why this one carries the topic.

Every practical situation in here becomes a graph, and the graph is the situation in disguise: the slope is a rate, the intercept is a starting amount, the line is what happens next. Read those parts back into the real thing — cost, distance, fuel — and the model tells you what you didn't measure.

An equal-scale coordinate plane with a green line y = mx + c whose gradient and intercept you set, both intercepts and a rise-over-run triangle marked, and a dashed amber partner that is parallel, perpendicular or a fixed line y = −x + 2 whose intersection is ringed and solved algebraically from the live values.

what examiners catch — In the assessment students swap m and c when substituting, read the x-intercept as c, or find x from the simultaneous pair and forget to substitute back for y. The mark comes from writing the two rules equal, solving for x, substituting to get y, and stating the solution as a point (x, y).
what you leave with

five things, not forty.

what's underneath

nothing here is a standalone fact.

knowhere maps every concept to what it rests on and what rests on it — 2 underneath this one, 4 built on top. Each one says why, in a sentence, not as an arrow on a diagram.

this conceptlinear relationshipsAn equal-scale coordinate plane with a green line y = mx + c whose gradient and intercept you set, both intercepts and a rise-over-run triangle marked, and a dashed amber partner that is parallel, perpendicular or a fixed line y = −x + 2 whose intersection is ringed and solved algebraically from the live values.
sits under itsolving linear equationsbecause a gradient and intercept are read off an equation you can already rearrange
sits under itlinear growth and decaybecause A constant rate of change per step is the gradient read as growth.
built on itleast squares regressionbecause Reading a and b in y = a + bx assumes the student knows what gradient and intercept are on a straight line.
built on itexponential and reciprocal relationshipsbecause the straight-line reading of gradient and intercept is the baseline these depart from
built on itnon-linear relationsbecause non-linear is defined against linear
built on itgraphs and models of practical situationsbecause reading a shape off a story starts with reading a straight line
the rest of algebra

2 more, same treatment.

Each one is its own knowscape in the app — built for how a particular student takes things in, not one explanation handed to everybody.

graphs and models of practical situationsin the appsimultaneous equations and the break-even pointin the app
this is one of 865

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knowhere›hsc mathematics standard 1›linear relationships