Solve the following system of equations for a and for b

Question

:
System of Equations:

Value of a

Value of b
9a+3b=30

8a+4b=28

Answer 1

To solve the system of equations:

1. $9a+3b=30$ (Equation 1)
2. $8a+4b=28$ (Equation 2)

We can simplify both equations.

Starting with Equation 1, we can divide every term by 3:

$$3a+b=10\text{(Equation 3)}$$

Now, let's simplify Equation 2 by dividing every term by 4:

$$2a+b=7\text{(Equation 4)}$$

Now we have the simplified system:

1. $3a+b=10$ (Equation 3)
2. $2a+b=7$ (Equation 4)

Next, we can eliminate $b$ by subtracting Equation 4 from Equation 3:

$$(3a+b)-(2a+b)=10-7$$

This simplifies to:

$$3a-2a+b-b=3$$

So, we have:

$$a=3$$

Now, we can substitute this value of $a$ back into either Equation 3 or Equation 4 to find $b$. Using Equation 4:

$$2(3)+b=7$$

This becomes:

$$6+b=7$$

Subtracting 6 from both sides gives:

$$b=1$$

Thus, the solution to the system of equations is:

$$a=3,b=1$$