Additional Mathematics Synthesis Guide 22: Simultaneous Equations, Line–Curve Intersections, Discriminants and Tangency Conditions

Two relationships → eliminate one variable → classify roots → recover points → interpret geometry → verify.

1. Simultaneous means both conditions must hold

If a point (x,y) lies on both y=x² and y=3x−2, then the same x and y must satisfy both equations. Equating the two y-expressions gives

x²=3x−2.

So

x²−3x+2=0=(x−1)(x−2).

Thus x=1 or 2. Returning to y=3x−2 gives the two intersection points

(1,1) and (2,4).

2. An x-root is not the whole intersection

The quadratic produced by substitution normally gives x-coordinates. A question asking for points requires corresponding y-values.

Using the simpler original equation for back-substitution usually reduces arithmetic. If y=3x−2 is available, it is cheaper than evaluating x² again.

Always check that each recovered point satisfies both original equations when the algebra is substantial.

3. Root count becomes intersection count

Suppose substituting a line into a parabola gives a quadratic

Ax²+Bx+C=0.

Then:

• Δ>0 → two distinct real x-values → usually two distinct intersection points.
• Δ=0 → one repeated x-value → tangent contact in the standard line-parabola setting.
• Δ<0 → no real x-values → no real intersection.

The geometry is being classified through the discriminant of the eliminated equation.

4. Worked secant example

Find the intersections of

y=x²−1

and

y=x+1.

Set equal:

x²−1=x+1.

So x²−x−2=0, giving x=2 or −1.

Then y=3 or 0, so the points are

(2,3) and (−1,0).

The discriminant is 9>0, consistent with two intersections.

5. Tangency as a repeated root

Find k so that

y=2x+k

is tangent to

y=x²+1.

Set equal:

x²+1=2x+k.

Thus x²−2x+1−k=0. For tangency,

Δ=4−4(1−k)=4k=0.

Hence k=0. The repeated root is x=1, giving tangent point (1,2).

6. Verify the tangent by differentiation

For y=x²+1, the derivative is 2x. At x=1, the gradient is 2, exactly the gradient of y=2x.

The point (1,2) lies on both the curve and line. This independent verification confirms that the discriminant-zero condition was applied to the correct simultaneous equation.

Discriminant and derivative routes are not competing facts; they are two mathematical descriptions of tangency.

7. A parameter can control whether graphs meet

For what k does the line

y=x+k

meet the parabola y=x²+2 at two distinct real points?

Equate:

x²+2=x+k

so x²−x+2−k=0. Two distinct intersections require

Δ=1−4(2−k)>0.

Thus 4k−7>0, giving

k>7/4.

At k=7/4 the line is tangent; below that value there is no real intersection.

8. Boundary values describe geometric transitions

In the previous family, k=7/4 is not an arbitrary algebraic threshold. It is the exact parameter at which the graph changes from no intersection to two intersections by passing through a one-contact tangent state.

This “phase change” interpretation is useful across parameter problems: the discriminant boundary often identifies where the number or type of real solutions changes.

9. Intersections with a horizontal line

For y=x²−4x+7 and horizontal line y=k, solve

x²−4x+7=k.

Completing the square gives

(x−2)²+3=k.

Therefore:

• k>3 → two intersections.
• k=3 → one tangent contact at the vertex.
• k<3 → no real intersection.

Here completed-square form reads the same classification more quickly than expanding a discriminant.

10. Simultaneous equations with a reciprocal curve

Find intersections of

y=6/x

and

y=x−1.

Since x≠0, set

6/x=x−1.

Multiply by x:

6=x²−x.

Thus x²−x−6=0, so x=3 or −2. Both are legal. The points are

(3,2) and (−2,−3).

The original domain restriction x≠0 remains active even though the transformed quadratic no longer displays a denominator.

11. A line and an exponential curve may not reduce to a quadratic

Not every line-curve intersection belongs to the discriminant method. For example, eˣ=x+3 does not become a polynomial quadratic through ordinary algebraic elimination.

The discriminant belongs only when the simultaneous system reduces to a quadratic in the chosen variable. Method selection begins by identifying the actual algebraic structure after elimination.

Graphical or numerical methods may be appropriate for other intersection families, depending on syllabus scope and the question.

12. Line-circle tangency uses the same repeated-root logic

Guide 20 showed that substituting a line into a circle produces a quadratic. Discriminant zero then means the line meets the circle at one repeated point, so it is tangent.

The reusable idea is therefore broader than “parabola tangent”: whenever an intersection problem reduces correctly to a quadratic and tangency means a double contact, Δ=0 is the algebraic signature.

13. Elimination instead of substitution

For some simultaneous systems, elimination is cleaner. Consider

x+y=5

and

x²+y²=13.

Square the linear relation:

x²+2xy+y²=25.

Since x²+y²=13, we get 2xy=12, so xy=6. Therefore x and y are the two roots of

t²−5t+6=0.

Hence {x,y}={2,3}, producing the ordered pairs (2,3) and (3,2).

14. Symmetry can reduce work

The previous system is symmetric in x and y. If (2,3) is a solution, swapping coordinates gives (3,2). Recognising symmetry prevents duplicate algebra and provides a verification expectation.

Symmetry should be inferred from the equations, not from a sketch that merely looks symmetric.

15. A system can produce extraneous candidates after squaring

If a simultaneous system includes a square root and we square both sides during elimination, the transformed equation may admit candidates that do not satisfy the original sign condition.

For example, solving y=√x and y=x−2 requires x≥0 and x−2≥0 at an intersection because √x is non-negative. Squaring x−2=√x gives

(x−2)²=x.

This transformed quadratic must be followed by substitution into the original system. The original equations remain the final authority.

16. The difference function packages an intersection problem

To find intersections of y=f(x) and y=g(x), define

h(x)=f(x)−g(x).

Then intersections occur where h(x)=0.

If h is quadratic, its discriminant classifies the intersection count. If h has a repeated zero, the graphs have equal y-values and often equal first-order contact at that x in a smooth line-parabola setting.

This viewpoint unifies “simultaneous equation” with ordinary root analysis.

17. Tangency from matching value and gradient

For smooth curves y=f(x) and y=g(x), a tangent contact at x=a normally satisfies

• f(a)=g(a) — same point;
• f′(a)=g′(a) — same tangent gradient.

For a straight line g(x)=mx+c and parabola f(x), these conditions reproduce the repeated-root tangency found by discriminant.

This gives a useful alternative when the tangent point is easier to parameterise than the line intercept.

18. Reverse problem: tangent line with unknown slope

Find tangent lines to y=x² passing through P=(0,−1).

A line through P has form

y=mx−1.

At intersection with y=x²:

x²=mx−1,

so x²−mx+1=0. Tangency requires

m²−4=0.

Thus m=±2. The tangent lines are

y=2x−1 and y=−2x−1.

The corresponding tangent x-values are 1 and −1.

19. A parameter can move a line without changing its slope

For y=mx+c with fixed m, varying c translates the line vertically. Relative to a fixed upward-opening parabola, there is often a critical c at which the line changes from missing the curve to touching it to cutting it twice.

The discriminant as a function of c reveals this threshold exactly. Geometrically, the family of parallel lines sweeps across the parabola.

20. Worked parallel-line family

For what c does y=4x+c intersect y=x² at two distinct real points?

Set x²=4x+c:

x²−4x−c=0.

Two distinct roots require

16+4c>0,

so

c>−4.

At c=−4, the line y=4x−4 is tangent at x=2, where y=4.

21. Common failure patterns

• Solving the eliminated equation and forgetting to recover the second coordinate.
• Using Δ=0 before correctly forming the quadratic that represents the intersection.
• Assuming every simultaneous system should be attacked by substitution even when elimination or symmetry is cleaner.
• Clearing denominators and forgetting original exclusions.
• Squaring during elimination and accepting every transformed root.
• Calling one intersection automatically tangent without checking the structure that produced the single solution.
• Using a discriminant method when the eliminated equation is not quadratic.
• Forgetting that parameter boundary values often belong to a different geometric case than strict inequalities.

22. A reliable intersection routine

1. Identify what the two equations represent and record domains.
2. Choose substitution, elimination or symmetry to reduce to one variable.
3. Determine the algebraic type of the reduced equation.
4. If it is quadratic, use the discriminant when root count is the target.
5. Solve for actual coordinates when required.
6. For tangency, verify the repeated point and, where useful, the matching gradient.
7. Return parameter boundary cases to their geometric interpretation.
8. Check final points in both original equations.

23. Practice set

1. Find intersections of y=x² and y=5x−6.
2. Find intersections of y=x²+1 and y=3x.
3. Classify the number of intersections of y=x²+4 and y=2x.
4. Find k so y=x+k is tangent to y=x².
5. Find the tangent point in Question 4.
6. For what k does y=x+k meet y=x² at two distinct points?
7. Find intersections of y=8/x and y=x−2.
8. Find intersections of x+y=7 and x²+y²=25.
9. Find tangent lines from (0,−4) to y=x².
10. For what c does y=6x+c meet y=x² in two distinct points?
11. Find c for tangency in Question 10.
12. Find the tangent point in Question 11.
13. Explain what Δ<0 means geometrically in a line-parabola simultaneous equation.
14. Explain what Δ=0 means.
15. Why must denominator restrictions survive after clearing fractions?
16. Why can squaring create extraneous intersection candidates?
17. State two conditions for smooth curves f and g to have tangent contact at x=a.
18. Define h(x)=f(x)−g(x). What do the zeros of h represent?
19. Why might elimination be preferable to substitution in a symmetric system?
20. Give one independent verification for a tangent line found by discriminant.

Answers

1. x²−5x+6=0, so x=2,3; points (2,4),(3,9).
2. x²−3x+1=0, so x=(3±√5)/2 and y=3x.
3. x²−2x+4=0 has Δ=−12, so no real intersection.
4. x²=x+k gives x²−x−k=0; tangency: 1+4k=0, so k=−1/4.
5. Repeated root x=1/2; y=1/4.
6. 1+4k>0, so k>−1/4.
7. 8/x=x−2 gives x²−2x−8=0, so x=4 or −2; points (4,2), (−2,−4).
8. (x+y)²=49 and x²+y²=25 give 2xy=24, so xy=12; t²−7t+12=0 gives 3,4; points (3,4),(4,3).
9. Line y=mx−4. Tangency with x² gives x²−mx+4=0, so m²−16=0; y=4x−4 and y=−4x−4.
10. x²−6x−c=0; Δ=36+4c>0, so c>−9.
11. c=−9.
12. x=3, y=9.
13. No real intersection.
14. One repeated intersection coordinate, giving tangent contact in the standard setting.
15. Because excluded values made the original expressions undefined even if the cleared equation later accepts them.
16. Squaring removes sign information, so a transformed equality may hold even when the original unsquared equality does not.
17. f(a)=g(a) and f′(a)=g′(a).
18. The x-coordinates where the graphs intersect.
19. It may exploit symmetric sums or cancellations and avoid unnecessary substitution expansion.
20. Check the tangent point lies on both graphs and that the line gradient matches the curve derivative there.

24. What mastery looks like

Mastery means the learner interprets simultaneous equations as shared constraints, selects the cheapest elimination route, uses discriminants only when the reduced equation is actually quadratic, understands parameter thresholds as changes in intersection geometry, and verifies tangency by point and gradient rather than treating Δ=0 as an isolated ritual.

The transfer test is to present the same problem as equations, as two graphs, and as a parameter question. If the learner can move between root count, intersection count and tangent conditions without losing domain restrictions, the simultaneous-equation system is connected.

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Continue through Batch 06

• Guide 21: Quadratic Functions, Completing the Square, Global Bounds and Modelling
• Guide 23: Binomial Theorem, General Terms, Coefficient Extraction and Parameter Problems
• Guide 24: Cubic Polynomials, Factor and Remainder Theorems, Roots, Multiplicity and Graph Structure
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