2x^3 + 3y^2 = 5z & 6x^3 + 7y^2 = 8z Add Both Equations By Substitution Method

To solve the system of equations using the substitution method, we need to solve one equation for a particular variable and substitute that into the other equation. Let's go step by step.

The system of equations is:

\[
1. \ 2x^3 + 3y^2 = 5z
\]
\[
2. \ 6x^3 + 7y^2 = 8z
\]

### Step 1: Solve the first equation for \(z\)
From the first equation:

\[
2x^3 + 3y^2 = 5z
\]

Solve for \(z\):

\[
z = \frac{2x^3 + 3y^2}{5}
\]

### Step 2: Substitute this expression for \(z\) in the second equation
Substitute \(z = \frac{2x^3 + 3y^2}{5}\) into the second equation \(6x^3 + 7y^2 = 8z\):

\[
6x^3 + 7y^2 = 8 \times \frac{2x^3 + 3y^2}{5}
\]

Now simplify the right-hand side:

\[
6x^3 + 7y^2 = \frac{8(2x^3 + 3y^2)}{5}
\]
\[
6x^3 + 7y^2 = \frac{16x^3 + 24y^2}{5}
\]

### Step 3: Eliminate the denominator
To eliminate the fraction, multiply both sides of the equation by 5:

\[
5(6x^3 + 7y^2) = 16x^3 + 24y^2
\]
\[
30x^3 + 35y^2 = 16x^3 + 24y^2
\]

### Step 4: Simplify the equation
Now, bring like terms together:

\[
30x^3 - 16x^3 = 24y^2 - 35y^2
\]
\[
14x^3 = -11y^2
\]

### Step 5: Express the relationship between \(x^3\) and \(y^2\)
Now, solve for \(x^3\) in terms of \(y^2\):

\[
x^3 = \frac{-11y^2}{14}
\]

### Step 6: Substitute \(x^3 = \frac{-11y^2}{14}\) into the equation for \(z\)
Now, substitute this into the expression for \(z = \frac{2x^3 + 3y^2}{5}\):

\[
z = \frac{2\left(\frac{-11y^2}{14}\right) + 3y^2}{5}
\]

Simplify:

\[
z = \frac{\frac{-22y^2}{14} + 3y^2}{5}
\]

Convert \(\frac{-22y^2}{14}\) to simplify:

\[
z = \frac{-\frac{11y^2}{7} + 3y^2}{5}
\]

Now combine the terms:

\[
z = \frac{-\frac{11y^2}{7} + \frac{21y^2}{7}}{5}
\]
\[
z = \frac{\frac{10y^2}{7}}{5}
\]

Finally:

\[
z = \frac{2y^2}{7}
\]

### Final Result
Thus, the relationship between \(x\), \(y\), and \(z\) is:

\[
x^3 = \frac{-11y^2}{14}, \quad z = \frac{2y^2}{7}
\]

Would you like further details or have any questions?

### Expanding Further:
1. How do you solve nonlinear equations using the elimination method?
2. What if we solve this system by graphing instead of substitution?
3. What would be the result if \(x = 1\) or \(y = 1\)?
4. How would you approach this problem with a different substitution variable?
5. What is the significance of having a negative term in these equations?

**Tip:** When solving by substitution, always ensure that your final expressions are fully simplified before concluding the solution.

Learn how to solve a nonlinear system of equations, including 2x^3 + 3y^2 = 5z and 6x^3 + 7y^2 = 8z, using the substitution method. This step-by-step solution explores algebraic techniques for solving nonlinear systems. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation