Answer
Here are the solutions for each equation with respect to \( x \):
1. \( bc = \frac{x^{2} - b^{2} - c^{2}}{2 \cos x} \)
No solution found for \( x \).
2. \( A = p(1+x)^{n} \)
\( x = \left(\frac{A}{p}\right)^{\frac{1}{n}} - 1 \)
3. \( mn = pn - x \)
\( x = -mn + np \)
4. \( y = ax^{2} \)
\( x = \frac{\sqrt{ay}}{a} \) or \( x = -\frac{\sqrt{ay}}{a} \)
5. \( y = mx + c \)
\( x = \frac{y - c}{m} \)
Solution
Solve the equation by following steps:
- step0: Solve for \(x\):
\(A=p\left(1+x\right)^{n}\)
- step1: Swap the sides:
\(p\left(1+x\right)^{n}=A\)
- step2: Divide both sides:
\(\frac{p\left(1+x\right)^{n}}{p}=\frac{A}{p}\)
- step3: Divide the numbers:
\(\left(1+x\right)^{n}=\frac{A}{p}\)
- step4: Calculate:
\(1+x=\left(\frac{A}{p}\right)^{\frac{1}{n}}\)
- step5: Move the constant to the right side:
\(x=\left(\frac{A}{p}\right)^{\frac{1}{n}}-1\)
Solve the equation \( y=mx+c \).
Solve the equation by following steps:
- step0: Solve for \(x\):
\(y=mx+c\)
- step1: Swap the sides:
\(mx+c=y\)
- step2: Move the expression to the right side:
\(mx=y-c\)
- step3: Divide both sides:
\(\frac{mx}{m}=\frac{y-c}{m}\)
- step4: Divide the numbers:
\(x=\frac{y-c}{m}\)
Solve the equation \( mn=pn-x \).
Solve the equation by following steps:
- step0: Solve for \(x\):
\(mn=pn-x\)
- step1: Swap the sides:
\(pn-x=mn\)
- step2: Move the expression to the right side:
\(-x=mn-pn\)
- step3: Change the signs:
\(x=-mn+pn\)
- step4: Simplify:
\(x=-mn+np\)
Solve the equation \( y=ax^{2} \).
Solve the equation by following steps:
- step0: Solve for \(x\):
\(y=ax^{2}\)
- step1: Swap the sides:
\(ax^{2}=y\)
- step2: Divide both sides:
\(\frac{ax^{2}}{a}=\frac{y}{a}\)
- step3: Divide the numbers:
\(x^{2}=\frac{y}{a}\)
- step4: Simplify the expression:
\(x=\pm \sqrt{\frac{y}{a}}\)
- step5: Simplify:
\(x=\pm \frac{\sqrt{ay}}{a}\)
- step6: Separate into possible cases:
\(\begin{align}&x=\frac{\sqrt{ay}}{a}\\&x=-\frac{\sqrt{ay}}{a}\end{align}\)
Here are the solutions for each equation with respect to \( x \):
1. **Equation:** \( bc = \frac{x^{2} - b^{2} - c^{2}}{2 \cos x} \)
**Solution:** No solution found for \( x \).
2. **Equation:** \( A = p(1+x)^{n} \)
**Solution:**
\[
x = \left(\frac{A}{p}\right)^{\frac{1}{n}} - 1
\]
3. **Equation:** \( mn = pn - x \)
**Solution:**
\[
x = -mn + np
\]
4. **Equation:** \( y = ax^{2} \)
**Solution:**
\[
x = \frac{\sqrt{ay}}{a} \quad \text{or} \quad x = -\frac{\sqrt{ay}}{a}
\]
5. **Equation:** \( y = mx + c \)
**Solution:**
\[
x = \frac{y - c}{m}
\]
If you need further assistance or explanations on any of these solutions, feel free to ask!
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