1.3 Solve the following equations simultaneously:
and

Ask by Cross Watkins. in South Africa

Mar 13,2025

Question

1.3 Solve the following equations simultaneously: $$ \sqrt{3^{x}} \cdot 9^{y}=27 \quad $$ and $$ \quad x+4 y^{2}=6 $$

UpStudy AI · Accepted Answer

We start with the given equations:

$$
\sqrt{3^{x}} \cdot 9^{y} = 27
$$
$$
x + 4 y^{2} = 6
$$

### Step 1. Rewrite the first equation

Notice that

$$
\sqrt{3^{x}} = 3^{\frac{x}{2}}
$$
and

$$
9^{y} = (3^2)^{y} = 3^{2y}.
$$

So the equation becomes

$$
3^{\frac{x}{2}} \cdot 3^{2y} = 27.
$$

Since multiplying powers with the same base adds exponents, we have

$$
3^{\frac{x}{2} + 2y} = 27.
$$

Recall that $$27 = 3^3$$. Hence, we equate the exponents:

$$
\frac{x}{2} + 2y = 3.
$$

Multiplying both sides by 2 to clear the fraction:

$$
x + 4y = 6.
$$

### Step 2. Form a new system of equations

Now we have the following two equations:

$$
x + 4y = 6 \quad (1)
$$
$$
x + 4y^{2} = 6 \quad (2)
$$

### Step 3. Subtract the equations to solve for $$y$$

Subtract equation (2) from equation (1):

$$
(x + 4y) - (x + 4y^2) = 6 - 6.
$$

This simplifies to:

$$
4y - 4y^2 = 0.
$$

Factor out $$4y$$:

$$
4y(1 - y) = 0.
$$

Thus, we have two cases:

1. $$4y = 0 \quad \Longrightarrow \quad y = 0$$,
2. $$1 - y = 0 \quad \Longrightarrow \quad y = 1$$.

### Step 4. Solve for $$x$$ in each case

**Case 1: $$y = 0$$**

Substitute $$y = 0$$ into equation (1):

$$
x + 4(0) = 6 \quad \Longrightarrow \quad x = 6.
$$

**Case 2: $$y = 1$$**

Substitute $$y = 1$$ into equation (1):

$$
x + 4(1) = 6 \quad \Longrightarrow \quad x = 2.
$$

### Step 5. Write the solutions

The simultaneous solutions are:

$$
(x, y) = (6, 0) \quad \text{or} \quad (x, y) = (2, 1).
$$
The solutions are $$ (x, y) = (6, 0) $$ or $$ (x, y) = (2, 1) $$.

1.3 Solve the following equations simultaneously:
and

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1.3 Solve the following equations simultaneously: and