Reading the Graph of Ax + By = C in Context
An equation like Ax + By = C graphs as a line whose intercepts show "spend everything on one item" and whose slope shows the trade-off rate between the two items.
Ax + By = C usually models a fixed total split between two things, like money spent on two ticket types or time spent on two tasks. The two intercepts are the extreme cases. Setting c = 0 means no child tickets at all, so the full budget C goes into adult tickets, and you solve A·a = C for the a-intercept. Setting a = 0 does the reverse. Every point between the intercepts is some mix of the two, but the ends tell you what happens when one quantity disappears entirely.
The slope tells you the rate of trade-off, not just a ratio of coefficients. Solve 7x + 4y = 56 for y: 4y = 56 - 7x, so y = 14 - (7/4)x. The slope is -7/4, meaning every time x goes up by 4, y drops by 7 to keep the total fixed. That negative sign is not optional decoration. It comes directly from moving the x term to the other side, and it reflects the fact that gaining more of one item costs you some of the other.
This reading connects straight to systems of equations, where two such lines meet at the one combination satisfying both constraints, and to inequalities, where the region on one side of the line represents every affordable or feasible mix rather than just the boundary itself.
Key terms
- intercept
- the point where the line crosses an axis, found by setting the other variable to zero.
- slope
- the rate at which y changes per unit change in x, equal to -A/B after solving Ax + By = C for y.
- standard form
- a linear equation written as Ax + By = C, with x and y terms on one side and a constant on the other.
- trade-off rate
- the real-world meaning of slope here: how much of one quantity you give up to gain a unit of the other.
Watch it explained
- How it worksHow do you find both intercepts of 3x + 5y = 60 by setting each variable to zero in turn?
- Worked exampleFind and interpret both intercepts of 4a + 10c = 200 where a and c count adult and child tickets.
- Worked exampleFind the slope of 7x + 4y = 56 and explain the trade it describes between x and y.
- Common mistakeWhy isn't the slope of Ax + By = C simply A divided by B?
Check yourself
End of topic.