Solve the simultaneous equations:
\[y = 2x\]
\[x + y = 6\]
One way to solve them is by using the substitution method.
Begin by labelling the equations (1) and (2):
\(y = 2x\)(1)
\(x + y = 6\)(2)
Equation (1) tells you that \(y = 2x\), so substitute this value of \(y\) into the second equation, ie replace \({y}\) with \({2x}\).
\[x + 2x = 6\]
\[3x = 6\]
\[x = 2\]
This gives you the value of \(x\), but what is the value of \(y\)?
Equation (1) tells you that \(y = 2x\), so \(y\) must be \(4\).
You can check your answer in the other equation - in this case, (2).
\[2 + 4 = 6\]
This is right, so the values are correct.
The solution of the equations is therefore:
\(x=2\), \(y=4\)
Use the substitution method to solve the following simultaneous equations:
\[y = x + 2\]
\[2x + y = 11\]
First label the equations.
\(y = x + 2\)(1)
\(2x + y = 11\)(2)
Substitute the value of \(y\) from equation (1) into equation (2):
\[2x + (x + 2) = 11\]
\[3x + 2 = 11\]
\[3x = 9\]
\[x = 3\]
To find \(y\) use this \(x\) value in equation (1): \(y = x + 2 = 3 + 2 = 5\)
Check in equation (2): \((2 \times 3) + 5 = 11\) (which is correct)
So the solution is:
\(x = 3\), \(y = 5\)