Problem 17 - Basic Engineering Mathematics

Question

Determine the values of \(x\) and \(y\) which simultaneously satisfy the equations: \(y=5 x-4-2 x^{2}\) and \(y=6 x-7\)

Step-by-Step Solution

Answer

\(x = -\frac{3}{2}, y = -16; x = 1, y = -1\).

Step 1: Equate the Expressions for y

We know that both equations express different relationships for \(y\). To find values of \(x\) where both equations are satisfied simultaneously, we set their right-hand sides equal to each other, giving the equation: \(5x - 4 - 2x^2 = 6x - 7\).

Step 2: Simplify the Equation

Rearrange the equation \(5x - 4 - 2x^2 = 6x - 7\) by moving all terms to one side: \(-2x^2 + 5x - 4 - 6x + 7 = 0\). Simplify to get \(-2x^2 - x + 3 = 0\).

Step 3: Solve the Quadratic Equation

Using the quadratic formula, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), with \(a = -2\), \(b = -1\), and \(c = 3\): calculate the discriminant \(b^2 - 4ac = (-1)^2 - 4(-2)(3) = 1 + 24 = 25\). The square root of the discriminant is 5. Substitute back into the formula: \[x = \frac{-(-1) \pm 5}{2(-2)} = \frac{1 \pm 5}{-4}\].

Step 4: Calculate the x-values

Compute the two possible values of \(x\): \(x = \frac{6}{-4} = -\frac{3}{2}\) and \(x = \frac{-4}{-4} = 1\).

Step 5: Find Corresponding y-values

For \(x = -\frac{3}{2}\), substitute into \(y = 6x - 7\): \(y = 6\left(-\frac{3}{2}\right) - 7 = -9 - 7 = -16\). For \(x = 1\), substitute into \(y = 6x - 7\): \(y = 6(1) - 7 = 6 - 7 = -1\).

Step 6: Conclusion: Solutions for x and y

The solutions for \(x\) and \(y\) that satisfy both equations are \((x, y) = \left(-\frac{3}{2}, -16\right)\) and \((x, y) = (1, -1)\).

Key Concepts

Simultaneous Equations

Simultaneous equations are a set of equations with multiple variables. We solve them to find a common solution for all the equations involved. The goal is to find the values of the variables that satisfy all the equations at the same time. In our given exercise, we have two equations representing lines in a coordinate plane.

• The first equation: \(y = 5x - 4 - 2x^2\), represents a quadratic curve.
• The second equation: \(y = 6x - 7\), represents a straight line.

To find where these two equations intersect, we set the expressions for \(y\) equal to each other, because, at the intersection points, they must share the same \(y\)-value. This leads us to the concept of equating their right-hand sides, which is the first step to finding solutions for both \(x\) and \(y\) simultaneously.

Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations, which are equations of the form \(ax^2 + bx + c = 0\). The formula is given by:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]This formula provides the solutions for \(x\) by substituting values for \(a\), \(b\), and \(c\) from the equation. In our context, the quadratic equation we derived from the simultaneous equations was \(-2x^2 - x + 3 = 0\).

• Here, the coefficients are \(a = -2\), \(b = -1\), and \(c = 3\).

Using the quadratic formula, we calculated the solutions for \(x\). Remember, these solutions also depend on another crucial component known as the discriminant, which influences the nature of the solutions.

Discriminant

The discriminant is a component of the quadratic formula, represented by \(b^2 - 4ac\). It helps in determining the nature of the roots of a quadratic equation. The discriminant is found under the square root in the quadratic formula.

• If \(b^2 - 4ac > 0\), there are two distinct real solutions.
• If \(b^2 - 4ac = 0\), there is exactly one real solution, indicating a repeated root.
• If \(b^2 - 4ac < 0\), the solutions are complex or imaginary.

In our problem, the discriminant was calculated as 25, which is positive. This indicates two distinct real solutions for \(x\). Having understood the discriminant, we can better predict the type of solutions to expect even before fully solving the quadratic equation.