Solve this system of equations \( \left\{\begin{array}{l}4 x+3 y=11 \\ 3 x-3 y=\quad-18\end{array}\right. \) One or more solutions: ONo solution O Infinite number of solutions

To solve the system of equations given by \( 4x + 3y = 11 \) and \( 3x - 3y = -18 \), we can start by manipulating the second equation for clarity. Dividing the second equation by 3 gives us \( x - y = -6 \), which simplifies our view of the relationship between \( x \) and \( y \). From this equation, we can express \( y \) in terms of \( x \): \( y = x + 6 \). Next, we substitute this expression for \( y \) into the first equation \( 4x + 3y = 11 \): \[ 4x + 3(x + 6) = 11 \] This expands to \( 4x + 3x + 18 = 11 \), leading to \( 7x + 18 = 11 \). Solving for \( x \), we find \( 7x = 11 - 18 \) or \( 7x = -7 \), thus \( x = -1 \). We plug \( x = -1 \) back into \( y = x + 6 \) to get \( y = -1 + 6 = 5 \). So we find a unique solution: \( (x, y) = (-1, 5) \). Therefore, there is one solution to the system. Now, you have a great foundation in solving systems of equations! To make your learning even more vibrant, you might want to check out some real-life applications: these systems actually model various scenarios, such as calculating costs or predicting outcomes in business or engineering! Moreover, it’s quite easy to mix-up signs in algebraic manipulations – be sure to double-check your calculations when rearranging or substituting! Clearly keeping track of your steps can help you avoid common errors and boost your confidence in solving equations effectively!