Sure, let's solve the simultaneous equations step by step.
We are given the system of equations:
[tex]\[ 2p + q = 2 \][/tex]
[tex]\[ p - q = -\frac{1}{2} \][/tex]
### Step 1: Solve one equation for one variable
First, let's solve the second equation for [tex]\( p \)[/tex]:
[tex]\[ p - q = -\frac{1}{2} \][/tex]
[tex]\[ p = q - \frac{1}{2} \][/tex]
### Step 2: Substitute the expression into the other equation
Now that we have [tex]\( p \)[/tex] in terms of [tex]\( q \)[/tex], substitute this expression into the first equation:
[tex]\[ 2p + q = 2 \][/tex]
[tex]\[ 2(q - \frac{1}{2}) + q = 2 \][/tex]
### Step 3: Simplify and solve for [tex]\( q \)[/tex]
Distribute and combine like terms:
[tex]\[ 2q - 1 + q = 2 \][/tex]
[tex]\[ 3q - 1 = 2 \][/tex]
Add 1 to both sides:
[tex]\[ 3q = 3 \][/tex]
Divide both sides by 3:
[tex]\[ q = 1 \][/tex]
### Step 4: Substitute [tex]\( q \)[/tex] back to find [tex]\( p \)[/tex]
Now, we substitute [tex]\( q = 1 \)[/tex] back into the expression we found for [tex]\( p \)[/tex]:
[tex]\[ p = q - \frac{1}{2} \][/tex]
[tex]\[ p = 1 - \frac{1}{2} \][/tex]
[tex]\[ p = 0.5 \][/tex]
### Final answer:
[tex]\[ p = 0.5 \][/tex]
[tex]\[ q = 1 \][/tex]
Thus, the values of [tex]\( p \)[/tex] and [tex]\( q \)[/tex] that satisfy the simultaneous equations are:
[tex]\[ p = 0.5 \][/tex]
[tex]\[ q = 1 \][/tex]
