Solutions For System Of Equations X - Y = 7 And Y = √(3x + 3) - 2

To determine the number of solutions for the given system of equations, we need to analyze the equations and find their points of intersection. The system of equations is:

1. x - y = 7
2. y = √(3x + 3) - 2

We will explore different methods to solve this problem, including algebraic manipulation and graphical analysis, to provide a comprehensive understanding.

Algebraic Approach

Step 1: Express x in terms of y from the first equation.

From the first equation, x - y = 7, we can express x as:

x = y + 7

This expression will be substituted into the second equation to solve for y.

Step 2: Substitute x in the second equation.

Substitute x = y + 7 into the second equation, y = √(3x + 3) - 2:

y = √(3(y + 7) + 3) - 2

This substitution results in a single equation in terms of y, which we can solve.

Step 3: Simplify and solve for y.

Simplify the equation:

y = √(3y + 21 + 3) - 2 y = √(3y + 24) - 2

To eliminate the square root, isolate the square root term and square both sides:

y + 2 = √(3y + 24) (y + 2)² = (√(3y + 24))² y² + 4y + 4 = 3y + 24

Now, rearrange the equation into a quadratic equation:

y² + 4y + 4 - 3y - 24 = 0 y² + y - 20 = 0

Step 4: Solve the quadratic equation.

We can solve the quadratic equation y² + y - 20 = 0 by factoring, using the quadratic formula, or completing the square. Factoring is often the simplest method if possible. We look for two numbers that multiply to -20 and add to 1. These numbers are 5 and -4.

So, we can factor the quadratic equation as:

(y + 5)(y - 4) = 0

This gives us two possible solutions for y:

y = -5 or y = 4

Step 5: Find the corresponding x values.

Using the expression x = y + 7, we find the corresponding x values for each y:

For y = -5: x = -5 + 7 = 2

For y = 4: x = 4 + 7 = 11

Thus, we have two potential solutions: (2, -5) and (11, 4).

Step 6: Check the solutions in the original equations.

It is crucial to check these solutions in the original equations because squaring both sides can introduce extraneous solutions. Let's check each potential solution:

1. For (2, -5):

   • Equation 1: x - y = 7 2 - (-5) = 2 + 5 = 7 (Correct)
   • Equation 2: y = √(3x + 3) - 2 -5 = √(3(2) + 3) - 2 -5 = √(6 + 3) - 2 -5 = √9 - 2 -5 = 3 - 2 -5 = 1 (Incorrect)

   Therefore, (2, -5) is not a solution.

2. For (11, 4):

   • Equation 1: x - y = 7 11 - 4 = 7 (Correct)
   • Equation 2: y = √(3x + 3) - 2 4 = √(3(11) + 3) - 2 4 = √(33 + 3) - 2 4 = √36 - 2 4 = 6 - 2 4 = 4 (Correct)

   Therefore, (11, 4) is a valid solution.

Conclusion from Algebraic Approach

From the algebraic approach, we found one valid solution: (11, 4). Thus, the system of equations has only one solution.

Graphical Approach

A graphical approach can provide a visual confirmation of the number of solutions. We will plot both equations on the same coordinate plane and observe their points of intersection.

Step 1: Rewrite the equations in a suitable form for plotting.

The first equation, x - y = 7, can be rewritten as y = x - 7. This is a linear equation with a slope of 1 and a y-intercept of -7.

The second equation, y = √(3x + 3) - 2, is a square root function. To plot this, we need to consider its domain and behavior.

Step 2: Analyze the domain of the square root function.

For the square root function y = √(3x + 3) - 2, the expression inside the square root must be non-negative:

3x + 3 ≥ 0 3x ≥ -3 x ≥ -1

Thus, the domain of the square root function is x ≥ -1.

Step 3: Plot the graphs of both equations.

1. Plot the line y = x - 7

   • We can find two points to plot this line easily. For example:
     • When x = 0, y = 0 - 7 = -7
     • When x = 7, y = 7 - 7 = 0
   • Plot the points (0, -7) and (7, 0) and draw a line through them.

2. Plot the square root function y = √(3x + 3) - 2

   • We know the domain is x ≥ -1. Let's find some points:
     • When x = -1, y = √(3(-1) + 3) - 2 = √0 - 2 = -2
     • When x = 0, y = √(3(0) + 3) - 2 = √3 - 2 ≈ 1.732 - 2 ≈ -0.268
     • When x = 1, y = √(3(1) + 3) - 2 = √6 - 2 ≈ 2.449 - 2 ≈ 0.449
     • When x = 11, y = √(3(11) + 3) - 2 = √36 - 2 = 6 - 2 = 4
   • Plot these points and sketch the curve of the square root function.

Step 4: Observe the points of intersection.

By plotting both graphs, we can observe that they intersect at only one point, which is (11, 4). This visually confirms our algebraic solution.

Conclusion from Graphical Approach

The graphical approach confirms that there is only one point of intersection between the two equations, reinforcing our conclusion that the system of equations has only one solution.

Final Answer

Combining both the algebraic and graphical approaches, we have definitively determined that the system of equations has only one solution. Therefore, the correct answer is:

B. 1

This comprehensive analysis ensures a clear understanding of the solution process and the underlying mathematical principles involved.