I just cant understand how author substite the second two equations into the first equation and get the system. Pls someone show it step by step.
$$Y=C+I+G$$ $$C=a+b(Y-T)$$ $$I=i_0-i_1r$$ $$M^s=c_1Y-c_2r$$
Author says:
We follow the standard method of substituting the second two equations into the first equation and simplifying to obtain the system.
and he get this $$(1-b)Y+i_1r=a+i_0+G-bT $$ $$c_1Y-c_2r=M^s$$
You just directly substitute the second and third equation into the first, and move things around: \begin{align} Y &= C + I + G \tag{given} \\ &= \left[ a + b(Y-T)\right] + \left[ i_0 - i_1r\right] + G \tag{substitution} \\ &= bY - i_1r + a + i_0 + G - bT \tag{basic rearranging} \end{align} Now, move the first two terms of the RHS to the LHS to get \begin{align} Y- bY + i_1r &= a + i_0 + G - bT \\ \end{align} Hence: \begin{align} (1- b)Y + i_1r &= a + i_0 + G - bT \end{align}
The equation $c_1Y-c_2r=M^s$ is exactly the fourth equation you're given