Simultaneous equations

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Notes de leçon

What simultaneous equations are

  • A simultaneous equations problem is a pair of linear equations that involve the same two variables, usually x and y.
  • A linear equation contains variables only to the first power, with no x2, xy or 1/x terms.
  • A solution is a pair of values (x, y) that makes both equations true at the same time.
  • Example: the pair 2x + 3y = 6 and 4x + 9y = 15 has the solution x = 3/2, y = 1.
  • Because both equations must hold at once, the solution is written as an ordered pair, e.g. (3/2, 1).

Solving by substitution

  • Substitution means rearranging one equation to make one variable the subject, then putting that expression into the other equation.
  • Example: from 2x + 3y = 6, rearrange to get x = 3 − (3/2)y.
  • Substitute that expression into the second equation: 4(3 − (3/2)y) + 9y = 15.
  • This leaves a single equation in one variable, which you solve in the usual way.
  • In the example, solving gives y = 1; then substitute back to find x = 3/2.
  • Always substitute back into one of the original equations to check both values work.

Solving by elimination

  • Elimination means adding or subtracting the two equations to remove one variable.
  • If the coefficients of one variable are the same, subtract the equations; if they are opposites, add them.
  • If the coefficients do not match, multiply one or both equations by a number to make them match first.
  • Example: 2x + 3y = 6 and 4x + 9y = 15 — multiply the first equation by 2 to get 4x + 6y = 12.
  • Subtracting the second equation from this gives −3y = −3, so y = 1.
  • Substitute y = 1 back to find x = 3/2, giving the same solution as substitution.

Solving graphically

  • Each linear equation is a straight line when drawn on a set of axes.
  • The solution of a pair of simultaneous equations is the point where the two lines cross.
  • Read the coordinates of the crossing point to find x and y.
  • If the lines are parallel they never cross, so there is no solution.
  • If the two equations describe the same line, every point on the line is a solution.

Forming equations from word problems

  • Choose a letter for each unknown quantity, such as p for the price of one item and q for the price of another.
  • Translate each sentence of the problem into an equation using those letters.
  • Example: if 2 pens and 3 pencils cost £6, write 2p + 3q = 6.
  • Use a second piece of information to write the second equation, e.g. 4p + 9q = 15.
  • Solve the pair by elimination or substitution, then state the answer in the context of the problem.
  • Check the answer makes sense — for example, prices should be positive.

Checking your solution

  • Substitute your values of x and y into both original equations.
  • Both equations must give true statements, e.g. 2(3/2) + 3(1) = 6 and 4(3/2) + 9(1) = 15.
  • If one equation fails, the solution is wrong — go back and check your working.
  • A quick sketch of the two lines can confirm the crossing point is in a sensible place.

Key vocabulary

  • Variable: a letter standing for an unknown number, such as x or y.
  • Coefficient: the number multiplying a variable, e.g. the 2 in 2x.
  • Linear system: a collection of two or more linear equations in the same variables.
  • Solution: the values of the variables that satisfy every equation at once.
  • Ordered pair: the solution written as (x, y).

Diapos

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  1. 1.In the pair of simultaneous equations 2x + 3y = 6 and 4x + 9y = 15, which variables are the unknowns?

    Easy
    • Ax and y
    • B2 and 3
    • C6 and 15
    • D4 and 9

  2. 2.The solution of a pair of simultaneous linear equations is a pair of values that satisfies both equations at the same time.

    Easy

    True or false?

  3. 3.Which of these is a linear equation?

    Medium
    • A3x + 2y = 7
    • Bx2 + y = 5
    • Cxy = 6
    • Dx2 + y2 = 9

  4. 4.On a graph, the solution of two simultaneous linear equations is found at which point?

    Medium
    • AWhere the two straight lines cross
    • BWhere each line meets the y-axis
    • CThe origin
    • DWhere each line meets the x-axis

  5. 5.The pair of equations 2x + 3y = 6 and 4x + 9y = 15 is solved by elimination. Which multiplier should be applied to the first equation so that the x terms cancel when the equations are added or subtracted?

    Medium
    • A2
    • B3
    • C4
    • D9

  6. 6.Solve the simultaneous equations 2x + 3y = 6 and 4x + 9y = 15. What is the value of x?

    Medium
    • A3/2
    • B1
    • C3
    • D6

  7. 7.Put the steps for solving 2x + 3y = 6 and 4x + 9y = 15 by substitution into the correct order.

    Medium
    • Substitute x = 3 - (3/2)y into 4x + 9y = 15 to get 4(3 - (3/2)y) + 9y = 15
    • Expand and solve to find y = 1
    • Substitute y = 1 back into x = 3 - (3/2)y to find x = 3/2
    • Check that x = 3/2 and y = 1 satisfy both original equations

  8. 8.Which of the following pairs of values are solutions to the simultaneous equations 2x + 3y = 6 and 4x + 9y = 15? (select all that apply)

    Hard
    • Ax = 1.5, y = 1
    • Bx = 3, y = 0
    • Cx = 0, y = 2
    • Dx = 3/2, y = 1
    • Ex = 4, y = -1

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