Simultaneous Equations Part C: Linear and Quadratic Solving!

Today I am going to do simultaneous equations again. So you know how to do the ‘elimination method’ and the ‘substitution method’ and you also know how to solve simultaneous equations when you are given a pair of linear equations but what about if you were given to solve a pair of simultaneous equations which has a linear equation and a quadratic equation?? What will you do?? Simple the way to solve it is exactly the same as how to solve a pair of linear equations simultaneously! The only catch is that we only use the substitution method. We can also use the ‘elimination method’ but it is much more feasible to use the ‘substitution method’. Let’s look at an example.

2x^2 + 3y^2 = 21……….eq’n 1
3x + 2y = 7………….…..eq’n 2

First step: Use the linear equation (eq’n 2) and make any unknown (x or y) the subject of the formula. Ok let’s do x first!

3x + 2y = 7
3x = 7 – 2y
x = (7 – 2y)/3

Ok let’s do y

3x + 2y = 7
2y = 7 – 3x
y = (7 – 3x)/2

For this question I will substitute y = (7 – 3x)/2 …..this is because you should always use the denominator with the lowest value.

Second step: sub. y = (7 – 3x)/2 into eq’n 1

2x^2 + 3y^2 = 21
2x^2 + 3( (7 – 3x)/2)^2 = 21
Do you remember how to expand (a - b)^2 ?? Well if you do expand it just like (7 – 3x)^2..but we are squaring (7 – 3x)/2……so jus square the top first to get …..49 – 42x + 9x^2 and now square the bottom……2^2 = 4.

Third step: re-write the question with the substitution of the unknown so that there is only one unknown in the quadratic equation.

2x^2 + 3( (49 – 42x + 9x^2)/4)) = 21

Now this can be solved just like a normal quadratic equation. REMEMBER YOU MUST GET 2 ANSWERS FOR X BECAUSE IT IS A QUADRATIC EQUATION!!!

By solving you get x = 3/5 and x = 3. Now you must put each value you got for x into the linear equation to solve for y.

When you do the substitution, you get y = 13/5 when x = 3/5 and y = -1 when x = 3.

NOTE: YOU GET 4 ANSWERS FOR A LINEAR AND A QUADRATIC AND 2 ANSWERS FOR A LINEAR AND A LINEAR

Believe me, the S.A.T questions are so different, but to do it you still the need to have the knowledge to do the questions. On my next post I will show you how the S.A.T brings their questions. They either give you it in written form which will be a long question to read or a visual form…we will look into more details in this, for now good luck learning simultaneous equations because that’s all to it.