Solve the following simultaneous equations. $$ \left\{\begin{array}{l}q=2 p^{3}-7 p+3 \ q=2 p^{3}-3 p-1\end{array} ight. $$ $$ p= $$ $$ q= $$

Question

Solve the following simultaneous equations. $$ \left\{\begin{array}{l}q=2 p^{3}-7 p+3 \ q=2 p^{3}-3 p-1\end{array}ight. $$ $$ p= $$ $$ q= $$

UpStudy AI · Accepted Answer

To solve the system of simultaneous equations:

$$
\begin{cases}
q = 2 p^{3} - 7 p + 3 \
q = 2 p^{3} - 3 p - 1
\end{cases}
$$

**Step 1: Equate the Two Expressions for $$ q $$**

Since both expressions equal $$ q $$, set them equal to each other:

$$
2 p^{3} - 7 p + 3 = 2 p^{3} - 3 p - 1
$$

**Step 2: Simplify the Equation**

Subtract $$ 2 p^{3} $$ from both sides:

$$
-7 p + 3 = -3 p - 1
$$

Bring all terms involving $$ p $$ to one side and constants to the other:

$$
-7 p + 3 + 3 p + 1 = 0 \
-4 p + 4 = 0
$$

**Step 3: Solve for $$ p $$**

$$
-4 p + 4 = 0 \
-4 p = -4 \
p = 1
$$

**Step 4: Substitute $$ p = 1 $$ Back into One of the Original Equations to Find $$ q $$**

Using the first equation:

$$
q = 2(1)^3 - 7(1) + 3 = 2 - 7 + 3 = -2
$$

**Solution:**

$$
\boxed{
\begin{aligned}
p &= 1 \
q &= -2
\end{aligned}
}
$$
$$ p = 1 $$ and $$ q = -2 $$.

Solve the following
simultaneous
equations.