Type the answers in the spaces provided. Use numbers instead of words. Determine the solution to the linear system by using the substitution method. \( \left\{\begin{array}{l}y=4 x+2 \\ y=3 x-2\end{array}\right. \) Answer Determine the solution to the linear system by using the substitution method. \( \left\{\begin{array}{l}3 x+2 y=4 \\ 2 x-y=5\end{array}\right. \) Answer (

To solve the first linear system using substitution, we can set the equations equal to each other since they both equal y: 1. \( 4x + 2 = 3x - 2 \) Subtract \( 3x \) from both sides: 2. \( 4x - 3x + 2 = -2 \) This simplifies to: 3. \( x + 2 = -2 \) Now, subtract 2 from both sides: 4. \( x = -4 \) Using the value of x in one of the equations to find y, we'll use \( y = 4x + 2 \): 5. \( y = 4(-4) + 2 \) 6. \( y = -16 + 2 = -14 \) So the solution to the first system is: \( (-4, -14) \) --- Now for the second system, we start with the first equation \( 3x + 2y = 4 \). We can isolate y: 1. \( 2y = 4 - 3x \) 2. \( y = \frac{4 - 3x}{2} \) Now we'll substitute this expression for y in the second equation \( 2x - y = 5 \): 3. \( 2x - \frac{4 - 3x}{2} = 5 \) To eliminate the fraction, multiply everything by 2: 4. \( 4x - (4 - 3x) = 10 \) 5. \( 4x - 4 + 3x = 10 \) Combining like terms gives us: 6. \( 7x - 4 = 10 \) Adding 4 to both sides: 7. \( 7x = 14 \) Dividing by 7: 8. \( x = 2 \) Now we substitute back to find y using \( y = \frac{4 - 3x}{2} \): 9. \( y = \frac{4 - 3(2)}{2} \) This simplifies to: 10. \( y = \frac{4 - 6}{2} = \frac{-2}{2} = -1 \) So the solution to the second system is: \( (2, -1) \)