Solving Equations with Technology

Have a go first. Then open the working one line at a time.

1. Match the y-terms

   (2) × 2: 4x − 2y = 2

   The y-terms are now +2y and −2y.

2. Add

   (1) + (2): 9x = 16

   The y-terms cancel.

3. Solve for x

   x = 16/9 = 1.78

   Keep the exact value for the next step.

4. Find y

   y = 2x − 1 = 23/9 = 2.56

   Put x into the second equation.

Have a go first. Then open the working one line at a time.

1. (a) Expand

   (2x + 3)(x − 5) = 2x² − 10x + 3x − 15 = 2x² − 7x − 15

   Multiply each term in the first bracket by each term in the second.

2. (a) One side zero

   2x² − 7x − 15 = 9 → 2x² − 7x − 24 = 0

   Subtract 9 from both sides. Here a = 2, b = −7, c = −24.

3. (b) Clear the fraction

   4x² + 7 = 9x

   Multiply every term by x, including the 9.

4. (b) One side zero

   4x² − 9x + 7 = 0

   Subtract 9x from both sides.

Have a go first. Then open the working one line at a time.

1. Standard form

   3x + 5y = 4, 7x − 2y = 23

   Variables on the left, numbers on the right, in the same order in both.

2. Solve

   Simultaneous equation solver: x = 3, y = −1

   Enter the coefficients row by row.

3. Check by graph

   Graph both lines: they meet at (3, −1)

   The solution of a 2 × 2 system is where the two lines cross.

Have a go first. Then open the working one line at a time.

1. Compare the left sides

   4x + 6y = 2(2x + 3y)

   The second left side is exactly twice the first.

2. Compare the constants

   2 × 7 = 14, but the second equation says 11

   The same expression cannot equal 14 and 11 at once.

3. Conclude

   No solution

   The lines have the same gradient and different intercepts. They are parallel and never meet.

4. On the calculator

   Simultaneous equation solver: it gives no values for x and y

   An error message instead of numbers means no solution. Do not change the equations to force an answer.

5. Part (b)

   4x + 6y = 14 is the same line as 2x + 3y = 7

   Every point on the line is a solution: infinitely many.

Have a go first. Then open the working one line at a time.

1. Standard form

   3x + y − 2z = 5, 2x + y − z = 4, x − 3y + 2z = −1

   Move each variable term to the left and each number to the right. Keep x, y, z in the same order in every equation.

2. Solve

   Simultaneous equation solver: x = 1.666…, y = 1.333…, z = 0.666…

   Enter three equations in three unknowns.

3. Exact values

   x = 5/3, y = 4/3, z = 2/3

   The decimals repeat, so write them as fractions.

Have a go first. Then open the working one line at a time.

1. Standard form

   2w + x − y + 3z = 3; w − 2x + 3y + z = 14; 3w + x + 2y − z = 10; −w + 4x + y + 2z = −1

   Rearrange the two equations that are not in standard form. Keep the order w, x, y, z.

2. Solve

   Simultaneous equation solver, 4 equations: w = 2, x = −1, y = 3, z = 1

   Choose 4 equations and 4 unknowns, then enter the 16 coefficients and 4 constants.

Have a go first. Then open the working one line at a time.

1. Define

   Let n = cost of a notebook ($), p = cost of a pen ($)

   Always say what the letters mean.

2. Equations

   3n + 5p = 9.70, 4n + 2p = 9.20

   One equation for each person.

3. Solve

   Simultaneous equation solver: n = 1.90, p = 0.80

   Use the solver, not guess and check.

4. Answer the question

   n + p = 1.90 + 0.80 = $2.70

   The question asks for the total, not for n or p.

Have a go first. Then open the working one line at a time.

1. The 4th term

   u₄ = u₁ + 3d = 14

   Use \(u_n=u_1+(n-1)d\).

2. The sum

   S₈ = (8/2)(2u₁ + 7d) = 124, so 2u₁ + 7d = 31

   Use \(S_n=\frac{n}{2}(2u_1+(n-1)d)\).

3. Solve

   Simultaneous equation solver: u₁ = 5, d = 3

   Two equations in two unknowns: u₁ + 3d = 14 and 2u₁ + 7d = 31.