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:

\[ 3a + b = 10 \quad \text{(Equation 3)} \]

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

\[ 2a + b = 7 \quad \text{(Equation 4)} \]

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:

\[ a = 3, \quad b = 1 \]