The cost of 5 party hats and 3 balloons is 35p. The cost of 3 party hats and 2 balloons is 22p. What is the cost of 8 party hats and 5 balloons? cost35p cost22p

## Step 1:<br />First, we need to set up the system of equations. Let's denote the cost of a party hat as \(x\) and the cost of a balloon as \(y\). From the problem, we can write the following equations:<br /><br />### \(5x + 3y = 35\)<br />### \(3x + 2y = 22\)<br /><br />## Step 2:<br />Next, we use the method of substitution or elimination to solve this system of equations. Here, we will use the method of elimination. <br /><br />To eliminate one of the variables, we can multiply the first equation by 2 and the second equation by 3, which gives us:<br /><br />### \(10x + 6y = 70\)<br />### \(9x + 6y = 66\)<br /><br />## Step 3:<br />Subtract the second equation from the first to eliminate \(y\):<br /><br />### \(10x - 9x = 70 - 66\)<br />### \(x = 4\)<br /><br />## Step 4:<br />Substitute \(x = 4\) into the second equation to solve for \(y\):<br /><br />### \(3x + 2y = 22\)<br />### \(3(4) + 2y = 22\)<br />### \(12 + 2y = 22\)<br />### \(2y = 10\)<br />### \(y = 5\)<br /><br />## Step 5:<br />Now that we have the cost of a party hat (\(x = 4\)) and the cost of a balloon (\(y = 5\)), we can calculate the cost of 8 party hats and 5 balloons:<br /><br />### \(8x + 5y = 8(4) + 5(5) = 32 + 25 = 57\)