2.8.10

Simultaneous Equations

Test yourself

Simultaneous Quadratic Equations

You can solve a pair of simultaneous equations where at least one contains a quadratic using the substitution method. This is where you rearrange one of the equations and substitute it into the other equation.

Illustrative background for ExampleIllustrative background for Example ?? "content

Example

  • Take the pair of simultaneous equations:
    • 2x2 + y = 5
    • 2x + y = 1
  • You can rearrange the second equation to get:
    • y = 1 − 2x
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Substitution

  • y = 1 − 2x can then be substituted in as an expression for y in
    2x2 + y = 5.
    • This gives 2x2 + 1 − 2x = 5
    • Which can be rearranged to 2x2 − 2x − 4 = 0
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Solving for x

  • You can then solve 2x2 − 2x − 4 = 0 to find values of x that satisfies the simultaneous equations.
    • 2x2 − 2x − 4 = 0
    • (2x + 2)(x − 2) = 0
  • Therefore x = −1 and x = 2
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Solving for y

  • You then need to find out the values of y by substituting in the values
    x = −1 and x = 2.
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Solving for y when x = −1

  • 2x + y = 1
  • 2(−1) + y = 1
  • y = 3
    • This means x = −1 and y = 3 is one solution to the simultaneous equations.
Illustrative background for Solving for y when x = 2Illustrative background for Solving for y when x = 2 ?? "content

Solving for y when x = 2

  • 2x + y = 1
  • 2(2) + y = 1
  • y = −3
    • This means x = 2 and y = −3 is the other solution to the simultaneous equations.

Jump to other topics

1Number

1.1Using Numbers

1.1.1Equal Symbols

1.1.2Negative Numbers

1.1.3BIDMAS

1.1.4Inverse Operations

1.1.5End of Topic Test- Ordering

1.1.6Factors

1.1.7Multiples

1.1.8Highest Common Factor

1.1.9Lowest Common Multiple

1.1.10Number of Ways to Do a Task

1.1.11Number of Combinations

1.1.12Number of Ways To Arrange Objects

1.1.13End of Topic Test - Factors, Multiples & Combos

1.1.14Standard Form

1.1.15Ordinary Numbers to Standard Form

1.1.16Addition in Standard Form

1.1.17Subtraction in Standard Form

1.1.18Multiplication in Standard Form

1.1.19Division in Standard Form

1.1.20Exam-Style Questions - Standard Form & Primes

1.1.21Long Division

1.1.22Prime Numbers

1.1.23End of Topic Test - Standard Form & Long Division

1.2Fractions, Decimals & Percentages

1.2.1Adding Fractions

1.2.2Subtracting Fractions

1.2.3Multiplying Fractions

1.2.4Dividing Fractions

1.2.5Mixed Fractions

1.2.6Calculating Percentages of Numbers

1.2.7Calculating Fractions of Numbers

1.2.8Multiply Decimals

1.2.9Place Value

1.2.10Recurring Decimals

1.2.11Exam-Style Questions - Recurring Decimals

1.2.12Convert Fractions & Terminating Decimals

1.2.13Convert Recurring Decimals

1.2.14Convert Decimals & Percentages

1.2.15Compare Decimals

1.2.16Compare Fractions

1.2.17Exam-Style Questions - Percentages

1.2.18End of Topic Test - Fractions & Decimals

1.3Powers & Roots

1.3.1Powers

1.3.2Roots

1.3.3Indices Advanced

1.3.4Estimating Powers

1.3.5Estimating Roots

1.3.6Multiplying Powers

1.3.7Dividing Powers

1.3.8Power Raised to Power

1.3.9Fractions Raised to a Power

1.3.10Numbers Raised to Negative Powers

1.3.11Roots of Numbers

1.3.12Numbers to the Power of 1 & 0

1.3.13Surds

1.3.14Simplify Surds

1.3.15Rationalise Denominator

1.3.16Exam-Style Questions - Indices

1.3.17End of Topic Test - Powers & Roots

1.4Accuracy

1.4.1Decimal Rounding

1.4.2Significant Figure Rounding

1.4.3Estimation

1.4.4Truncation

1.4.5Limits of Accuracy

1.4.6Bounds Addition

1.4.7Bounds Subtraction

1.4.8Bounds Multiplication

1.4.9Bounds Division

1.4.10End of Topic Test - Rounding & Estimation

2Algebra

2.1Introduction to Algebra

2.1.1Algebra Basics

2.1.2Substituting Values

2.1.3Algebraic Vocabulary

2.1.4Collecting Terms

2.1.5End of Topic Test - Algebra Basics & Power Rules

2.1.6Expanding Brackets I

2.1.7Common Factors

2.1.8Expanding Brackets II

2.1.9Rearrange For x

2.1.10Rearrange with Powers

2.1.11Exam-Style Questions - Basics of Algebra

2.1.12End of Topic Test - Expanding & Rearranging

2.2Manipulating Algebra

2.2.1How to Factorise

2.2.2Factorising Quadratics

2.2.3Algebraic Fractions - Addition & Subtraction

2.2.4Algebraic Fractions - Multiplication & Division

2.3Proofs & Functions

2.3.1Identities & Proofs

2.3.2Exam-Style Questions - Algebraic Proof

2.3.3End of Topic Test - Algebra & Proofs

2.3.4Functions

2.3.5Combine Functions

2.3.6Composite Functions

2.3.7Inverse Functions

2.4Straight Line Graphs

2.4.1Straight Line Graphs

2.4.2y = mx + c

2.4.3Grad + YIntercept

2.4.4Exam-Style Questions - Straight Line Graphs

2.4.5End of Topic Test - Functions & Straight Lines

2.5Common Graphs

2.5.1Shape of Quadratic Graphs

2.5.2Roots of a Quadratic Graph

2.5.3Completing Squares

2.5.4Turning Points

2.5.5Common Functions 1

2.5.6Common Functions 2

2.5.7Trigonometric functions

2.5.8End of Topic Test - Common Graphs

2.6Transformations & Tangents

2.6.1Transformations

2.6.2Tangent on a Circle

2.7Properties of Graphs

2.7.1Estimate Gradient

2.7.2Estimate Area

2.7.3Represent Gradient & Area

2.7.4Distance-Time Graph

2.7.5Velocity-Time Graph

2.7.6End of Topic Test - Transformations & Graphs

2.8Solving Equations

2.8.1Solve Linear Equations

2.8.2Linear Equations with Fractions

2.8.3Solve Linear Equations Graphically

2.8.4Quadratic ±

2.8.5Factorise Quadratics

2.8.6Quadratic Formula

2.8.7Complete the Square

2.8.8Solve Quadratics Graphically

2.8.9End of Topic Test - Solving Equations

2.8.10Simultaneous Equations

2.8.11Quadratic Simultaneous Equations

2.8.12Simultaneous Equations - Graphs

2.8.13Simultaneous Equations - Word Problems

2.8.14Approximate Solutions - Number of Iterations

2.8.15Approximate Solutions - Iterations to 1 DP

2.8.16Approximate Solutions - Iteration Formula

2.8.17Exam-Style Questions - Simultaneous Equations

2.8.18End of Topic Test - Simultaneous Equations

2.9Inequalities

2.9.1Solving Inequalities

2.9.2Solving Double Inequalities

2.9.3Solving Quadratic Inequalities

2.9.4Representing Inequalities

2.9.5Exam-Style Questions - Inequalities

2.9.6End of Topic Test - Inequalities

2.10Sequences

2.10.1Nth Term Rules

2.10.2Recurrence Relationships

2.10.3Find Nth Term in Linear Sequence

2.10.4Check If a Number is In a Sequence

2.10.5Exam-Style Questions - Sequences

2.10.6Arithmetic & Geometric Progressions

2.10.7Key Sequences

2.10.8Quadratic Sequences

2.10.9Find Nth Term in Quadratic Sequence

2.10.10Exam-Style Questions - Geometric Series

2.10.11End of Topic Test - Sequences

3Ratio

3.1Ratios in Practice

3.1.1Converting Between Units

3.1.2Formula Triangles

3.1.3Scale Factors

3.1.4Ratios in Shapes

3.2Manipulating Ratios

3.2.1Simplifying Ratios

3.2.2Turn Ratios into Equations

3.2.3Ratio Notation & Reduction

3.2.4End of Topic Test - Units & Ratios

3.2.5Part : Part Ratio

3.2.6Part : Whole Ratio

3.2.7Two Ratios

3.2.8Differences in Ratios

3.2.9Fractions in Ratios

3.2.10Numbers as Fractions

3.2.11Ratios to Fractions

3.2.12Exam-Style Questions - Ratios

3.2.13End of Topic Test - Fractions & Ratios

3.3Percentage & Interest

3.3.1Convert Percentages to Decimals

3.3.2Convert Percentages to Fractions

3.3.3Find the Percentage

3.3.4More/Less Percentage

3.3.5Price from Percentage Change

3.3.6Calculate Percentage Change from Price

3.3.7End of Topic Test - Percentages

3.3.8Profit & Interest & Discount

3.3.9Original Price after Discounts

3.3.10Simple Interest

3.3.11Compound Growth & Decay

3.3.12Compound Interest

3.3.13Compound Interest - After How Many Days?

3.3.14End of Topic Test - Percentages in Finance

3.4Proportion

3.4.1Proportion

3.4.2Direct Proportion

3.4.3Inverse Proportion

3.4.4Exam-Style Questions - Proportionality

3.5Gradient

3.5.1Gradient on a Straight Line

3.5.2Direct & Inverse Proportions as Graphs

3.5.3Gradient of a Curve

3.5.4Positive & Negative Gradients

3.5.5End of Topic Test - Proportion & Gradient

4Geometry

4.1Introduction to Geometry

4.1.1Geometry Basics

4.1.2Geometry Basics 2

4.1.3Regular Polygons

4.1.4Triangles & Angles

4.1.5Midpoint

4.1.6Bisectors & Perpendiculars

4.1.7Locus

4.1.8Angle rules

4.1.9Parallel Angle Rules

4.1.10Angles in a Polygon

4.1.11Angles in a Triangle

4.1.12Exam-Style Questions - Angles in Shapes

4.1.13End of Topic Test - Geometry

4.2Triangles & Quadrilaterals

4.2.1Types of Triangle

4.2.2Angles & Triangles

4.2.3Congruence

4.2.4Similarity

4.2.5Quadrilaterals

4.3Transformations

4.3.1Types of Transformations

4.3.2Invariant Points

4.3.3Effects of Transformations

4.3.4Exam-Style Questions - Reflection

4.4Circle Basics

4.4.1Properties of a Circle 1

4.4.2Chords

4.4.3Properties of a Circle 2

4.4.4Area of a Circle

4.4.5Circumference of a Circle

4.4.6Fractions of Circles

4.4.7Exam-Style Questions - Circle Formulae

4.4.8End of Topic Test - Triangles & Circles

4.5Circle Theorems

4.5.1Perpendicular Bisector of a Chord

4.5.2Angle at the Centre

4.5.3Angle of a Semicircle

4.5.4Opposite Angles in Cyclic Quadrilaterals

4.5.5Tangent Lengths

4.5.6Angle in the Same Segment

4.5.7Alternate Segment Theorem

4.5.8Exam-Style Questions - Circle Theorems

4.6Measurements & Units

4.6.1Length

4.6.2Area

4.6.3Volume

4.6.4Capacity

4.6.5Time

4.6.6Money

4.6.7Mass

4.6.8Bearings

4.6.9Scale Factor & Enlargement

4.6.10Exam-Style Questions - Enlargements

4.7Calculating Area

4.7.1Area of 2D Shapes

4.7.2Exam-Style Questions - Area

4.7.3End of Topic Test - Circle Theorems, Units & Area

4.8Triangle Formulae

4.8.1Pythagoras' Theorem

4.8.2Pythagoras in 3D

4.8.3Trigonometry

4.8.4Exact Values

4.8.5Sine Rule

4.8.6Cosine Rule

4.8.7Area of a Triangle

4.8.8Exam-Style Questions - Trigonometry & Area

4.93D Shapes

4.9.1Types of 3D Shape

4.9.2Plans & Elevations

4.9.3Nets

4.9.4Volume of 3D Shapes

4.9.5Volume of a Cuboid

4.9.6Volume of a Cylinder

4.9.7Volume of a Sphere

4.9.8Volume of a Hemisphere

4.9.9Volume of a Cone

4.9.10Volume of a Pyramid

4.9.11Surface Area & Composite Shapes

4.9.12Surface Area of a Cuboid

4.9.13Surface Area of a Cylinder

4.9.14Surface Area of a Sphere/Hemisphere

4.9.15Surface Area of a Cone

4.9.16Exam-Style Questions - Volume

4.10Vectors

4.10.1Vectors Introduction

4.10.2Vectors

4.10.3Translation

4.10.4Scalar

4.10.5Exam-Style Questions - Vectors

4.10.6End of Topic Test - Trigonometry, Shapes & Vectors

5Probability

5.1Introduction to Probability

5.1.1Randomness & Fairness

5.1.2Sum of Probabilities

5.1.3Relative & Expected Frequency

5.1.4Frequency Tables

5.1.5Using Frequency Trees

5.1.6Complete Frequency Trees

5.2Representing Probabilities

5.2.1Probability

5.2.2Probability Scale

5.2.3Sample Space Diagrams

5.2.4Sets

5.2.5Venn Diagrams

5.2.6Probabilities in Venn Diagrams

5.2.7Exam-Style Questions - Venn Diagrams

5.3Probability Calculations

5.3.1Theoretical Probabilities

5.3.2Multiple Events

5.3.3Probability Calculations

5.3.4Dependent Events

5.3.5Independent Events

5.3.6Probability Tree Diagram

5.3.7Complex Tree Diagrams

5.3.8Conditional Probability

5.3.9Conditional Probability From a Two Way Table

5.3.10Conditional Probability Venn Diagrams

5.3.11Conditional Probability Tree Diagrams

5.3.12Exam-Style Questions - Probability

5.3.13End of Topic Test - Probability

6Statistics

6.1Introduction to Statistics

6.1.1Statistics Basics

6.1.2Statistics Key Terms

6.1.3Frequency Tables

6.1.4Exam-Style Questions - Statistics & Probability

6.2Charts & Diagrams

6.2.1Bar Charts

6.2.2Pie Charts

6.2.3Stem & Leaf Diagrams

6.2.4Back to Back Stem & Leaf Diagrams

6.2.5End of Topic Test - Statistics

6.2.6Pictograms

6.2.7Time Series Graphs

6.2.8Histograms

6.2.9Histograms 2

6.2.10Frequency Polygons

6.2.11Cumulative Frequency

6.2.12Cumulative Frequency 2

6.2.13Scatter Graphs

6.2.14Scatter Graphs - Predictions

6.2.15Correlation & Causation

6.2.16Exam-Style Questions - Cumulative Frequency

6.3Comparing Datasets

6.3.1Comparing Datasets

6.3.2End of Topic - Datasets

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End of Topic Test - Solving Equations

Quadratic Simultaneous Equations