How many solutions does the system of equations $x-y=7$ and $y=\sqrt{3 x+3}-2$ have?
A. 0
B. 1
C. 2
D. Infinitely many
Answer (1)
Solve the first equation for x : x = y + 7 .
Substitute into the second equation and simplify to get y + 2 = 3 y + 24 .
Square both sides and rearrange to form a quadratic equation: y 2 + y − 20 = 0 .
Solve the quadratic equation to find y = − 5 or y = 4 , and check the solutions in the original equations, resulting in one valid solution. Thus, the system has 1 solution.
Explanation
Problem Analysis We are given a system of two equations: x − y = 7 and y = 3 x + 3 − 2 . Our goal is to find the number of solutions to this system.
Solving for x First, let's solve the first equation for x : x = y + 7 .
Substitution Now, substitute this expression for x into the second equation: y = 3 ( y + 7 ) + 3 − 2 .
Simplifying the Equation Simplify the equation: y = 3 y + 21 + 3 − 2 , which becomes y = 3 y + 24 − 2 .
Isolating the Square Root Isolate the square root: y + 2 = 3 y + 24 .
Squaring Both Sides Square both sides of the equation: ( y + 2 ) 2 = 3 y + 24 , which expands to y 2 + 4 y + 4 = 3 y + 24 .
Rearranging into Quadratic Form Rearrange the equation into a quadratic equation: y 2 + y − 20 = 0 .
Solving the Quadratic Equation Solve the quadratic equation for y . This can be done by factoring: ( y + 5 ) ( y − 4 ) = 0 , so y = − 5 or y = 4 .
Finding Corresponding x Values For each value of y , find the corresponding value of x using x = y + 7 . If y = − 5 , then x = − 5 + 7 = 2 . If y = 4 , then x = 4 + 7 = 11 . So we have two potential solutions: ( 2 , − 5 ) and ( 11 , 4 ) .
Checking the Solutions Now, we need to check both solutions in the original equations to make sure they are valid.
Checking (2, -5) For ( 2 , − 5 ) : x − y = 2 − ( − 5 ) = 7 , which is true. However, y = 3 x + 3 − 2 gives − 5 = 3 ( 2 ) + 3 − 2 = 9 − 2 = 3 − 2 = 1 , which is false. Therefore, ( 2 , − 5 ) is not a solution.
Checking (11, 4) For ( 11 , 4 ) : x − y = 11 − 4 = 7 , which is true. Also, y = 3 x + 3 − 2 gives 4 = 3 ( 11 ) + 3 − 2 = 36 − 2 = 6 − 2 = 4 , which is true. Therefore, ( 11 , 4 ) is a solution.
Conclusion Since only one solution satisfies both equations, the system has one solution.
Examples
Systems of equations are used in various real-world applications, such as determining the break-even point for a business. For example, if a company's cost function is C ( x ) = 5 x + 1000 and its revenue function is R ( x ) = 15 x , where x is the number of units sold, we can set up a system of equations to find the break-even point where cost equals revenue: 5 x + 1000 = 15 x . Solving this system gives the number of units the company needs to sell to cover its costs. Similarly, the problem we solved involves finding the intersection of a line and a square root function, which can model various physical phenomena.
