Solving Pairs Of Equations A Detailed Guide With Examples

Understanding the Equations

The given pair of equations presents an interesting challenge. We have a linear equation, $y = 20 - 2x$, and a quadratic equation, $y = x^2 - 16x + 68$. The linear equation represents a straight line, while the quadratic equation represents a parabola. The solutions to this pair of equations are the points where these two graphs intersect. These points satisfy both equations simultaneously, making them the common solutions. To find these points, we need to employ algebraic manipulation and potentially numerical approximation techniques.

The Linear Equation: y = 20 - 2x

This equation is in slope-intercept form ($y = mx + b$), where $m$ is the slope and $b$ is the y-intercept. In this case, the slope is -2, indicating that the line slopes downward from left to right, and the y-intercept is 20, meaning the line crosses the y-axis at the point (0, 20). Linear equations are straightforward to graph and manipulate, making them a fundamental concept in algebra. Understanding the slope and y-intercept allows us to quickly visualize the line's behavior and its position on the coordinate plane. This understanding is crucial when solving systems of equations, as it helps us anticipate where the line might intersect with other curves, such as parabolas.

The Quadratic Equation: y = x² - 16x + 68

This equation is a quadratic equation in the standard form ($y = ax^2 + bx + c$), where $a = 1$, $b = -16$, and $c = 68$. Quadratic equations form parabolas when graphed. The coefficient 'a' determines whether the parabola opens upwards (if a > 0) or downwards (if a < 0). In this case, since $a = 1$, the parabola opens upwards. The vertex of the parabola is the point where it changes direction, and its x-coordinate can be found using the formula $x = -b / 2a$. In this case, the x-coordinate of the vertex is $-(-16) / (2 * 1) = 8$. Substituting $x = 8$ into the equation gives us the y-coordinate of the vertex: $y = 8^2 - 16 * 8 + 68 = 4$. Thus, the vertex of the parabola is (8, 4). Knowing the vertex and the direction the parabola opens helps us sketch its graph and understand its behavior, which is essential for finding its intersection points with the linear equation.

Solving the Equations Algebraically

To solve the system of equations, we can use the substitution method. Since both equations are expressed in terms of $y$, we can set them equal to each other:

$20 - 2x = x^2 - 16x + 68$

Now, we rearrange the equation to form a standard quadratic equation:

$0 = x^2 - 14x + 48$

This is a quadratic equation in the form $ax^2 + bx + c = 0$, where $a = 1$, $b = -14$, and $c = 48$. We can attempt to factor this quadratic equation, or if factoring is not straightforward, we can use the quadratic formula.

Factoring the Quadratic Equation

We look for two numbers that multiply to 48 and add up to -14. These numbers are -6 and -8. Therefore, we can factor the quadratic equation as:

$(x - 6)(x - 8) = 0$

Setting each factor equal to zero gives us the solutions for $x$:

$x - 6 = 0$ or $x - 8 = 0$

So, $x = 6$ or $x = 8$.

Finding the Corresponding y Values

Now that we have the $x$ values, we can substitute them back into either of the original equations to find the corresponding $y$ values. Let's use the linear equation $y = 20 - 2x$:

For $x = 6$:

$y = 20 - 2(6) = 20 - 12 = 8$

For $x = 8$:

$y = 20 - 2(8) = 20 - 16 = 4$

Thus, the solutions to the system of equations are $(6, 8)$ and $(8, 4)$. These are the points where the line and the parabola intersect.

Cases Requiring Approximation to Two Decimal Places

In some instances, the quadratic equation we obtain may not be factorable, or the solutions may involve irrational numbers. In such cases, we need to use the quadratic formula to find the solutions for $x$:

$x = (-b ± √(b^2 - 4ac)) / 2a$

If the discriminant ($b^2 - 4ac$) is not a perfect square, the solutions for $x$ will be irrational numbers. We then use a calculator to approximate these solutions to two decimal places.

The Quadratic Formula

For the general quadratic equation $ax^2 + bx + c = 0$, the quadratic formula provides a direct method for finding the solutions for $x$. The quadratic formula is particularly useful when the equation cannot be easily factored. It involves substituting the coefficients $a$, $b$, and $c$ into the formula and simplifying. The expression inside the square root, $b^2 - 4ac$, is known as the discriminant. The discriminant's value determines the nature of the roots: if it's positive, there are two distinct real roots; if it's zero, there is one real root (a repeated root); and if it's negative, there are two complex roots. Understanding the discriminant helps us anticipate the type of solutions we will obtain before fully applying the quadratic formula.

Approximating Solutions to Two Decimal Places

When the solutions for $x$ are irrational numbers, we need to approximate them to a specified number of decimal places. This involves using a calculator to compute the square root and perform the necessary arithmetic operations. Approximating to two decimal places means rounding the result to the nearest hundredth. This level of precision is often sufficient for practical applications, providing a balance between accuracy and simplicity. After finding the approximate values for $x$, we substitute them back into one of the original equations to find the corresponding approximate values for $y$. These approximate solutions represent the intersection points of the curves with a high degree of accuracy.

Example with Approximation

Let's consider a hypothetical scenario where solving a pair of equations leads to the quadratic equation:

$x^2 - 5x + 3 = 0$

This equation does not factor easily, so we use the quadratic formula:

$x = (5 ± √((-5)^2 - 4 * 1 * 3)) / (2 * 1)$

$x = (5 ± √(25 - 12)) / 2$

$x = (5 ± √13) / 2$

Here, $√13$ is an irrational number, so we approximate it using a calculator:

$√13 ≈ 3.61$

Therefore, the solutions for $x$ are:

$x ≈ (5 + 3.61) / 2 ≈ 4.31$

$x ≈ (5 - 3.61) / 2 ≈ 0.69$

Now, we substitute these approximate $x$ values into one of the original equations to find the corresponding $y$ values. For instance, if one of the original equations was $y = 2x - 1$, we would have:

For $x ≈ 4.31$:

$y ≈ 2(4.31) - 1 ≈ 7.62$

For $x ≈ 0.69$:

$y ≈ 2(0.69) - 1 ≈ 0.38$

So, the solutions to two decimal places are approximately $(4.31, 7.62)$ and $(0.69, 0.38)$.

Graphical Interpretation

Graphically, solving a pair of equations means finding the points where the graphs of the equations intersect. For a linear and a quadratic equation, there can be zero, one, or two intersection points. The graphical interpretation provides a visual confirmation of the algebraic solutions. When we solve the equations algebraically and find the solutions, these points should correspond to the intersection points on the graph. If the graphs do not intersect, it means there are no real solutions to the system of equations. If the line is tangent to the parabola, there is exactly one solution. Two intersection points indicate two distinct solutions.

Visualizing the Solutions

Plotting the graphs of the equations can help us visualize the solutions. We can use graphing software or online tools to plot the line and the parabola. Visualizing the solutions allows us to see how the algebraic solutions correspond to the intersection points on the graph. It also provides a way to check the accuracy of our algebraic solutions. If the calculated solutions do not match the intersection points on the graph, it indicates an error in our calculations. Graphing the equations is a valuable tool for understanding and verifying the solutions to a system of equations.

Conclusion

Solving pairs of equations, especially when approximations to two decimal places are necessary, requires a combination of algebraic skills and numerical methods. Mastering these techniques is essential for various mathematical and real-world applications. Whether through factoring, using the quadratic formula, or employing graphical methods, the ability to accurately solve equations is a cornerstone of problem-solving in mathematics and related fields.