Find the value of a, b, c, d.

Steps to Solve

1. Write down the equations

The given equations are:

1. $2a + b = 20$ (Equation 1)
2. $5c - d = 4$ (Equation 2)
3. $a - 2b = -34$ (Equation 3)
4. $4c + 3d = 24$ (Equation 4)

2. Solve for $b$ using Equation 1

Rearranging Equation 1 gives us:

$$ b = 20 - 2a $$

Now we can substitute this expression for $b$ into Equation 3.

3. Substitute $b$ in Equation 3

Substituting $b$ into Equation 3:

$$ a - 2(20 - 2a) = -34 $$

Simplifying it:

$$ a - 40 + 4a = -34 $$ $$ 5a - 40 = -34 $$ $$ 5a = 6 $$ $$ a = \frac{6}{5} \quad (\text{or } 1.2) $$

4. Find $b$

Now substitute $a$ back into the expression for $b$:

$$ b = 20 - 2(1.2) $$ $$ b = 20 - 2.4 $$ $$ b = 17.6 $$

5. Solve for $c$ and $d$ using Equations 2 and 4

Next, we need to solve the equations for $c$ and $d$.

First, from Equation 2, rearranging gives:

$$ d = 5c - 4 $$

Now substituting into Equation 4:

$$ 4c + 3(5c - 4) = 24 $$ $$ 4c + 15c - 12 = 24 $$ $$ 19c - 12 = 24 $$ $$ 19c = 36 $$ $$ c = \frac{36}{19} \quad (\text{or approximately } 1.8947) $$

6. Find $d$

Now substitute $c$ back into the expression for $d$:

$$ d = 5\left(\frac{36}{19}\right) - 4 $$ $$ d = \frac{180}{19} - \frac{76}{19} $$ $$ d = \frac{104}{19} \quad (\text{or approximately } 5.4737) $$

7. Final Values

The final values are:

• ( a = \frac{6}{5} )
• ( b = 17.6 )
• ( c = \frac{36}{19} )
• ( d = \frac{104}{19} )