VT · Simultaneous Equations (Linear & Quadratic)

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In each set, one thing changes and everything else stays the same. Work them out in order and look for the pattern — the last line tells you what to notice.

①The parabola stays $y=x^2$; the line changeschanging: the line

Solve each pair simultaneously. Give the coordinates of every intersection point.

$y=x+2$

$y=2x$

$y=3x-2$

$y=4x-4$

The last line meets the curve only ONCE – it is a tangent. What is special about its equation once you substitute?

②The circle stays $x^2+y^2=25$; the line changeschanging: the line

Solve each pair simultaneously. Give the coordinates of every intersection point.

$y=x+1$

$y=x-1$

$y=7-x$

$x=3$

Every answer point lies on the circle of radius 5. Substitute each one back to check $x^2+y^2=25$.

③The line stays $y=2x$; the parabola changeschanging: the parabola

Solve each pair simultaneously. Give the coordinates of every intersection point.

$y=x^2$

$y=x^2-3$

$y=x^2-8$

Only the constant term of the parabola changes. Watch how the two solutions spread apart as it drops.

Answers · Simultaneous Equations (Linear & Quadratic)

Variation practice

① The parabola stays $y=x^2$; the line changes

$y=x+2$: (2, 4) and (-1, 1)$y=2x$: (0, 0) and (2, 4)$y=3x-2$: (1, 1) and (2, 4)$y=4x-4$: (2, 4) only (tangent)

② The circle stays $x^2+y^2=25$; the line changes

$y=x+1$: (3, 4) and (-4, -3)$y=x-1$: (4, 3) and (-3, -4)$y=7-x$: (3, 4) and (4, 3)$x=3$: (3, 4) and (3, -4)

③ The line stays $y=2x$; the parabola changes

$y=x^2$: (0, 0) and (2, 4)$y=x^2-3$: (-1, -2) and (3, 6)$y=x^2-8$: (-2, -4) and (4, 8)