Problem 23 - Beginning and Intermediate Algebra

Question

Solve each system. $$\begin{array}{c} x^{2}+y^{2}=3 \\ x+y=4 \end{array}$$

Step-by-Step Solution

Answer

The given system of equations is: \[ \begin{cases} x^2 + y^2 = 3 \\ x + y = 4 \end{cases} \] Solving for y in the second equation, we get \(y = 4 - x\). Substituting this into the first equation and simplifying, we obtain the quadratic equation \(2x^2 - 8x + 13 = 0\). Using the quadratic formula, we find that there are no real solutions for x, as there is a negative value inside the square root. Therefore, the circle and line in our system do not intersect, and there are no common solutions for this system of equations.

Step 1: Solve for y in the second equation

We begin by solving for y in the second equation, x+y=4. Subtract x from both sides to find an expression for y: \(y = 4 - x\).

Step 2: Substitute the value of y in the first equation

Now that we have an expression for y, we can substitute it into the first equation: \(x^2 + y^2 = 3\). Replace y with \(4 - x\): \(x^2 + (4 - x)^2 = 3\).

Step 3: Simplify and solve the equation for x

Now we have an equation with just x. Expand and simplify the equation: \(x^2 + (4 - x)^2 = x^2 + (16 - 8x + x^2) = 3\). Combine like terms: \(2x^2 - 8x + 13 = 0\). This is a quadratic equation, and we can solve for x using either factoring, completing the square, or the quadratic formula. In this case, the quadratic formula is the quickest method.

Step 4: Use the quadratic formula to solve for x

The quadratic formula is given by: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). In our equation \(2x^2 - 8x + 13 = 0\), a = 2, b = -8, and c = 13. Plug in these values: \(x = \frac{8 \pm \sqrt{(-8)^2 - 4(2)(13)}}{2(2)} = \frac{8 \pm \sqrt{64 - 104}}{4} = \frac{8 \pm \sqrt{-40}}{4}\). Since there is a negative value inside the square root, there are no real solutions for x.

Step 5: Conclusion

Because there are no real solutions for x in the quadratic equation, it means that the circle and line in our system do not intersect. Therefore, there are no common solutions for this system of equations.

Key Concepts

Quadratic Equations

Quadratic equations are mathematical expressions that involve variables raised to the power of two. Typically, they take the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a \) is not zero. This form is known as the standard form of a quadratic equation.

Quadratic equations can be solved using different methods, including:

• Factoring
• Completing the square
• Using the quadratic formula

Understanding the characteristics of quadratic equations is essential for solving problems involving curves or parabolas. The equation in our example, \( 2x^2 - 8x + 13 = 0 \), is already arranged in standard form, which makes it convenient for identifying the coefficients needed for the substitution method and quadratic formula.

Substitution Method

The substitution method is a technique used to solve systems of equations where one equation is used to express one variable in terms of the other. This substitution is then used in the second equation to find the solution. This method is particularly useful when dealing with simultaneous equations, where at least one equation is linear.

In the given exercise, we solve the linear equation \( x + y = 4 \) for \( y \), finding that \( y = 4 - x \). This expression gets substituted into the quadratic equation \( x^2 + y^2 = 3 \).

This substitution simplifies the first equation to involve only one variable, making it easier to solve. Although it didn't lead to a real solution in this scenario, substitution remains a vital tool for solving systems of equations.

Quadratic Formula

The quadratic formula is a versatile tool for finding solutions to quadratic equations when other methods, like factoring, are not feasible. The formula is:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Here, \( a \), \( b \), and \( c \) correspond to the coefficients of the quadratic equation in standard form \( ax^2 + bx + c = 0 \).

In the solution of our system, the equation \( 2x^2 - 8x + 13 = 0 \) is solved for \( x \) using the quadratic formula. Plugging the relevant values into the formula reveals a negative number under the square root. This indicates an important concept related to solutions: when the discriminant, \( b^2 - 4ac \), is negative, the quadratic equation has no real roots.

No Real Solutions

The scenario with a negative discriminant highlights a significant aspect of quadratic equations—when \( b^2 - 4ac < 0 \), the solutions are not real numbers but rather involve imaginary numbers. This reflects that the graph of the equation (as part of a system involving a circle and a line) does not intersect at any real point.

Such situations are common when solving systems involving a circle \( x^2 + y^2 = 3 \) and a line \( x + y = 4 \) that don't cross one another. Recognizing when a system has no real solutions is crucial for interpreting mathematical models and verifying the steps leading to that conclusion. It's an essential skill that helps in knowing when a system of equations may only have solutions in the complex number system, highlighting the importance of understanding both the real and imaginary planes.