A system of equations is given. Equation 1:5x-2y=10 Equation $$ 2: 4 x-3 y=1^{15} $$ Explain how to eliminate $$ x $$ in the system of equations.

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To eliminate the variable $$ x $$ from the given system of equations, you can use the **method of elimination**. Here's a step-by-step explanation of how to achieve this:

### Given System of Equations:
1. **Equation 1:** $$ 5x - 2y = 10 $$
2. **Equation 2:** $$ 4x - 3y = 1^{15} $$

*Note:* $$ 1^{15} = 1 $$, so Equation 2 simplifies to:
$$ 4x - 3y = 1 $$

### Step 1: Make the Coefficients of $$ x $$ Identical

To eliminate $$ x $$, we need the coefficients of $$ x $$ in both equations to be the same (in absolute value). We can achieve this by finding the Least Common Multiple (LCM) of the coefficients of $$ x $$ in both equations.

- **Coefficient of $$ x $$ in Equation 1:** 5
- **Coefficient of $$ x $$ in Equation 2:** 4
- **LCM of 5 and 4:** 20

**Multiply each equation by the necessary factor to make the coefficient of $$ x $$ equal to 20:**

- **Multiply Equation 1 by 4:**
  $$
  4 \times (5x - 2y) = 4 \times 10 \
  \Rightarrow 20x - 8y = 40 \quad \text{(Equation 3)}
  $$

- **Multiply Equation 2 by 5:**
  $$
  5 \times (4x - 3y) = 5 \times 1 \
  \Rightarrow 20x - 15y = 5 \quad \text{(Equation 4)}
  $$

### Step 2: Subtract One Equation from the Other to Eliminate $$ x $$

Now that both Equation 3 and Equation 4 have the same coefficient for $$ x $$, subtract one equation from the other to eliminate $$ x $$.

- **Subtract Equation 4 from Equation 3:**
  $$
  (20x - 8y) - (20x - 15y) = 40 - 5 \
  \Rightarrow 20x - 8y - 20x + 15y = 35 \
  \Rightarrow 7y = 35
  $$

- **Solve for $$ y $$:**
  $$
  y = \frac{35}{7} = 5
  $$

### Summary

By multiplying the first equation by 4 and the second equation by 5, both equations have the same coefficient for $$ x $$ (which is 20). Subtracting one from the other eliminates $$ x $$, allowing you to solve for $$ y $$. Once $$ y $$ is found, you can substitute it back into either original equation to find the value of $$ x $$.

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**Final Answer:**  
Multiply the first equation by 4 and the second by 5 so both have 20x. Then subtract one equation from the other, canceling out the x terms. This eliminates x from the system.
To eliminate $$ x $$, multiply the first equation by 4 and the second by 5 to make the coefficients of $$ x $$ equal. Then subtract the second equation from the first to cancel $$ x $$, solving for $$ y $$.

A system of equations is given. Equation 1:5x-2y=10 Equation \( 2: 4 x-3 y=1^{15} \) Explain how to eliminate \( x \) in the system of equations.

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