Algebra - Simultaneous equations - Equating coefficients.
Test Yourself 1 - Solutions.
Solve the following equations simultaneously by first equating the coefficients for one variable.
Note if we are given an equation with a variable having a coefficient of 1, we would simply revert to our previous technique.
2x + 3y = 19 3x + 2y = 16 To equate the y coefficients multiply Eqn 1 by 2 and Eqn 2 by 3 4x + 6y = 38 9x + 6y = 48 5x = 10 x = 2 2(2) + 3y = 19 3y = 15 y = 5 So x = 2 and y = 5 |
3x - 5y = 11 2x - 3y = 8 To equate the y coefficients multiply Eqn 1 by 3 and Eqn 2 by 5 9x - 15y = 33 10x - 15y = 40 x = 7 10(7) - 15y = 40 15y = 30 y = 2 So x = 7 and y = 2.
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12y + 7z - 13 = 0 Eqn 2 multiplied by 2 becomes 12y + 2z + 22 = 0 Subtracting the modified Eqn 2 from Eqn 1 gives 5z = 35 z = 7 6y + 7 + 11 = 0 6y = -18 y = -3 So y = -3 and z = 7
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7a - 3h = 9 5a + 2h = 23 To equate the h coefficients (which have different signs) multiply Eqn 1 by 2 and Eqn 2 by 3 14a - 6h = 18 15a + 6h = 69 29a = 87 a = 3 5(3) + 2h = 23 2h = 8 h = 4 So a = 3 and h = 4. |
3m + 4n = -5 4m - 3n = 10 To equate the n coefficients (which have different signs) multiply Eqn 1 by 3 and Eqn 2 by 4 9m + 12n = -15 16m - 12n = 40 25m = 25 m = 1 9(1) + 12n = -15 12n = -24 n = -2 So m = 1 and n = -2 |
3c + 4d = 16 7c - 2d = 60 To equate the d coefficients (which have different signs) multiply Eqn 2 by 2 and then add Eqn 1 to Eqn 2: 14c - 4d = 120 17c = 136 c = 8 3(8) + 4d = 16 4d = -8 d = -2 So c = 8 and d = -2 |
Criterion:
If you attained at least 4 correct, you have made a good start. Re-read the solutions and review at least parts of Video 3 again. Then re-do this test.
If you attained, 5 correct, you have established a good basis for this topic. Compare your solutions with those above. The point where there is a difference highlights where you are making the mistake. Make a note of mistake in your revision notes so, through review, you will remember not to make that error again.
Again, talk to yourself about what you should be doing at each step so as to keep your concentration high and to ensure you take the right action.