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.

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 \( x \). --- **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.