Solve simultaneous equations linear and quadratic

Algebra • Solving equations and inequalities

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Result of substituting a linear expression into a quadratic

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A quadratic equation in one variable whose solutions give x-values for intersections.

Key concepts

What you'll likely be quizzed about

Definition and purpose of simultaneous equations

Simultaneous equations are two equations in the same two variables that must be satisfied at the same time. The intersection of their solution sets produces one or more ordered pairs (x, y) that satisfy both equations. Solutions represent points where both algebraic conditions hold simultaneously. Limitations arise when equations are dependent (infinite solutions) or inconsistent (no solution). A pair of identical linear equations produces infinitely many solutions. Parallel linear equations produce no common solution.

Algebraic method: substitution

Substitution isolates one variable in one equation and substitutes that expression into the other equation. Substitution converts a system into a single equation in one variable and therefore reduces the problem dimension. Substitution produces exact values when algebraic manipulation leads to solvable equations. When a linear/quadratic pair exists, substitution of the linear expression into the quadratic produces a quadratic equation. Solutions of the quadratic correspond to x-values of intersection; corresponding y-values result from back-substitution. Careful simplification and factorisation or the quadratic formula produce exact roots.

Algebraic method: elimination (addition/subtraction)

Elimination combines the two equations to remove one variable by adding or subtracting suitable multiples. Elimination produces a single linear equation when applied to two linear equations, allowing direct solution for one variable. The chosen multiple causes one variable's coefficients to cancel, producing a simpler equation. Elimination requires coefficient alignment and occasional scaling by constants. Elimination produces exact answers efficiently for linear/linear systems and avoids solving intermediate quadratics. Verification by substitution confirms the final pair of values.

Solving linear/quadratic systems algebraically

A linear/quadratic system produces up to two intersection points because a line can meet a parabola at zero, one, or two points. Substitution of the linear expression into the quadratic produces a quadratic equation whose discriminant determines the number of real intersections. A positive discriminant produces two distinct real solutions, zero produces one (tangent), and negative produces no real intersections. Algebraic solution requires solving the resulting quadratic by factorisation, completing the square, or the quadratic formula and then calculating the matching y-values using the linear equation.

Graphical method and approximation

Graphical solution plots both equations on the same coordinate axes and identifies intersection coordinates as simultaneous solutions. Graphs provide approximate values when exact algebra is complex or when a quick estimate is acceptable. Precision depends on scale, resolution and plotting accuracy. Graphical reading requires attention to intercepts and the scale on each axis. For quadratic/linear intersections, a clear plotting of the parabola and the line clarifies whether intersections exist and provides approximate x and y values to one or two decimal places as required.

Verification and limitations

Every algebraic or graphical solution requires verification by substituting the pair into both original equations to confirm equality. Graphical approximations require rounding and therefore demand statement of approximation. Algebraic methods produce exact rational or irrational answers, whereas graphical methods produce approximate decimal values only. Limitations include rounding error, plotting resolution and algebraic mistakes. The discriminant limits the number of real solutions in linear/quadratic pairs and determines whether algebraic real solutions exist.

Key notes

Important points to keep in mind

Substitute when one variable is already isolated or easily isolated.

Use elimination when coefficients can be scaled to cancel a variable cleanly.

After substitution into a quadratic, use factorisation or the quadratic formula to find exact x-values.

Use the discriminant to determine the number of real intersections in linear/quadratic systems.

Plot both equations accurately and read intersections to estimate solutions when algebraic work is difficult.

State graphical answers as approximations and include the degree of rounding.

Always substitute final (x, y) pairs back into both original equations to verify correctness.

Clear fractions early to reduce arithmetic mistakes during algebraic manipulation.

Parallel linear equations indicate no solution; identical linear equations indicate infinitely many solutions.

One-variable equations arising from elimination or substitution must be solved before finding the matched second variable.