What is the point of inte \( \begin{array}{l}y=3 x-7 \\ 2 x+y=13\end{array} \)

It looks like you're asking for the **point of intersection** of the two linear equations: \[ \begin{align*} y &= 3x - 7 \quad \text{(Equation 1)} \\ 2x + y &= 13 \quad \text{(Equation 2)} \end{align*} \] To find the **point of intersection**, we need to find the values of \( x \) and \( y \) that satisfy both equations simultaneously. Here's a step-by-step method to solve this system of equations: ### Method 1: Substitution 1. **Express one variable in terms of the other:** Equation 1 already expresses \( y \) in terms of \( x \): \[ y = 3x - 7 \] 2. **Substitute this expression into the second equation:** Replace \( y \) in Equation 2 with \( 3x - 7 \): \[ 2x + (3x - 7) = 13 \] 3. **Combine like terms and solve for \( x \):** \[ 2x + 3x - 7 = 13 \\ 5x - 7 = 13 \\ 5x = 13 + 7 \\ 5x = 20 \\ x = \frac{20}{5} \\ x = 4 \] 4. **Find \( y \) using the value of \( x \) in Equation 1:** \[ y = 3(4) - 7 \\ y = 12 - 7 \\ y = 5 \] ### Method 2: Elimination Alternatively, you can use the elimination method: 1. **Write both equations:** \[ \begin{align*} y &= 3x - 7 \quad \text{(Equation 1)} \\ 2x + y &= 13 \quad \text{(Equation 2)} \end{align*} \] 2. **Subtract Equation 1 from Equation 2:** \[ (2x + y) - y = 13 - (3x - 7) \\ 2x = 13 - 3x + 7 \\ 2x + 3x = 20 \\ 5x = 20 \\ x = 4 \] 3. **Find \( y \) using the value of \( x \) in Equation 1:** \[ y = 3(4) - 7 = 12 - 7 = 5 \] ### **Solution** Both methods lead to the same solution. **The point of intersection is at \((4,\,5)\).** ### **Verification** To ensure that this point lies on both lines: 1. **Check in Equation 1:** \[ y = 3(4) - 7 = 12 - 7 = 5 \quad \checkmark \] 2. **Check in Equation 2:** \[ 2(4) + 5 = 8 + 5 = 13 \quad \checkmark \] Since \((4, 5)\) satisfies both equations, it's indeed the correct point of intersection.