Question

Find the values of the constants m and b, for which y=mx + b is a solution to this equation y'=1/2x + y - 1

Answer 1

You need to find values of \(m \text{ and }b\) that satisfy the equation
\[m=\frac{1}{2}x+mx+b-1 \implies x(\frac{1}{2}+m)+b-1=m\]
Now, equate coefficients, you get the system:
\[\frac{1}{2}+m=0 \implies m=-\frac{1}{2}\]
\[b-1=m \implies b=m+1=\frac{1}{2}.\]
So, \(m=-\frac{1}{2}\) and \(b=\frac{1}{2}\).