How do you determine if a point is a solution to a system of equations, what does it mean if a system has no solution, and what are the two numbers if their sum is 25 and the larger is one more than t

Step 1: Answer Question 5 by substituting the given point into the equations. Given the system of equations: 1) 3x - 2y - 9 = 0 2) x + 8y - 3 = 0 To check if (3, 0) is a solution, substitute x=3 and y=0 into each equation. For equation 1: 3(3) - 2(0) - 9 = 9 - 0 - 9 = 0 This equation holds true. For equation 2: 3 + 8(0) - 3 = 3 + 0 - 3 = 0 This equation also holds true. Since (3, 0) satisfies both equations, it is a solution. Answer: Yes Step 2: Answer Question 6. If a system of equations has no solution, it means the lines represented by the equations never intersect. This occurs when the lines are parallel and distinct. Answer: parallel Step 3: Answer Question 7 by setting up and solving a system of equations. Let the smaller number be s and the larger number be l. From the problem statement, we have two conditions: 1) The sum of two numbers is 25: s + l = 25 2) The larger number is one more than twice the smaller number: l = 2s + 1 Substitute the expression for l from the second equation into the first equation: s + (2s + 1) = 25 3s + 1 = 25 Subtract 1 from both sides: 3s = 24 Divide by 3: s = 8 Now substitute the value of s back into the equation for l: l = 2(8) + 1 l = 16 + 1 l = 17 The larger number is 17. Answer: 17 Step 4: Answer Question 8 by solving the system of simultaneous equations. Given the system of equations: 1) 3a + 5b = 7 2) a - 2b = 4 From equation (2), express a in terms of b: a = 4 + 2b Substitute this expression for a into equation (1): 3(4 + 2b) + 5b = 7 12 + 6b + 5b = 7 12 + 11b = 7 Subtract 12 from both sides: 11b = 7 - 12 11b = -5 b = -(5)/(11) Now substitute the value of b back into the expression for a: a = 4 + 2(-(5)/(11)) a = 4 - (10)/(11) To subtract, find a common denominator: a = (4 × 11)/(11) - (10)/(11) a = (44)/(11) - (10)/(11) a = (34)/(11) The solution (a, b) is ((34)/(11), -(5)/(11)). Answer: ((34)/(11), -(5)/(11)) 3 done, 2 left today. You're making progress.