Linear Equations Finding Gradients, Y-Intercepts, And Solving For Variables
In the realm of mathematics, understanding the equation of a straight line is fundamental. The equation provides a concise way to describe the relationship between two variables and allows us to analyze the line's properties, such as its slope and points of intersection with the axes. In this article, we will delve into the equation of a straight line, explore how to determine its gradient and y-intercept, and work through practical examples to solidify our understanding. We will also address solving for variables in linear equations. This exploration is crucial for students, educators, and anyone seeking a deeper understanding of basic algebraic concepts. A solid grasp of these concepts forms the bedrock for more advanced mathematical studies, such as calculus and linear algebra. By mastering the fundamentals of linear equations, we not only enhance our mathematical abilities but also develop critical thinking and problem-solving skills applicable across various disciplines.
Question 56: Unveiling the Gradient and Y-Intercept of a Line
To truly grasp the essence of a straight line equation, let's dissect a classic problem. The equation of a straight line is given by $3x - 2y - 6 = 0$. Our mission is twofold: first, to find the gradient of this line, and second, to determine its $y$-intercept. These two elements, the gradient and the $y$-intercept, paint a comprehensive picture of the line's orientation and position on the coordinate plane. The gradient, often denoted as 'm', signifies the steepness and direction of the line. A positive gradient indicates an upward slope, while a negative gradient indicates a downward slope. The magnitude of the gradient reflects how steeply the line rises or falls. The $y$-intercept, on the other hand, is the point where the line crosses the $y$-axis. This point is crucial as it provides a fixed reference on the vertical axis, allowing us to visualize the line's vertical position. By finding both the gradient and the $y$-intercept, we gain a complete understanding of the line's behavior and its relationship with the coordinate system.
i) Determining the Gradient
The gradient of a line, often denoted by m, is a measure of its steepness and direction. It tells us how much the $y$-value changes for every unit change in the $x$-value. To find the gradient, we need to rearrange the given equation into the slope-intercept form, which is $y = mx + c$, where m is the gradient and c is the $y$-intercept. Let's start with the equation $3x - 2y - 6 = 0$. The first step is to isolate the term with $y$. We can do this by adding $2y$ to both sides of the equation, giving us $3x - 6 = 2y$. Now, to get $y$ by itself, we divide both sides of the equation by 2, resulting in $\frac{3}{2}x - 3 = y$. By simply rearranging, we've transformed the equation into the familiar slope-intercept form. From this form, we can directly identify the gradient as the coefficient of $x$. In this case, the coefficient of $x$ is $\frac{3}{2}$. Therefore, the gradient of the line is $\frac{3}{2}$. This positive gradient indicates that the line slopes upwards from left to right. The magnitude of $\frac{3}{2}$ tells us that for every 1 unit increase in $x$, the $y$-value increases by 1.5 units.
ii) Identifying the Y-Intercept
The y-intercept is the point where the line crosses the $y$-axis. At this point, the $x$-coordinate is always 0. To find the $y$-intercept, we look at the slope-intercept form of the equation, which we derived earlier: $y = \frac{3}{2}x - 3$. In this form, the constant term, c, represents the $y$-intercept. In our equation, the constant term is -3. Therefore, the $y$-intercept is -3. This means the line intersects the $y$-axis at the point (0, -3). We can also find the $y$-intercept by substituting $x = 0$ into the original equation: $3(0) - 2y - 6 = 0$. This simplifies to $-2y - 6 = 0$. Adding 6 to both sides gives us $-2y = 6$. Dividing both sides by -2, we get $y = -3$, confirming our earlier result. Understanding the $y$-intercept is crucial for visualizing the line's position on the coordinate plane. It provides a fixed point of reference, allowing us to accurately sketch the line and understand its relationship with the axes.
Question 57: Solving for Y in a Linear Equation
Stepping beyond the geometrical interpretation of lines, let's tackle a different type of problem: solving for a variable in a linear equation. Consider the equation $11y = (18)^2 - (15)^2$. Our objective here is to find the value of $y$. This type of problem emphasizes the importance of algebraic manipulation and the order of operations. We need to simplify the equation step-by-step, isolating $y$ on one side to determine its value. This process involves performing arithmetic operations, such as squaring, subtraction, and division, while maintaining the equation's balance. Mastery of these techniques is crucial for solving more complex algebraic problems and for applying mathematical principles to real-world scenarios. Solving for variables is a fundamental skill that underpins various mathematical disciplines, including calculus, statistics, and linear algebra.
To find the value of $y$, we need to simplify the equation step by step. Let's start with the given equation: $11y = (18)^2 - (15)^2$. First, we need to evaluate the squares. $18^2$ is 18 multiplied by itself, which equals 324. Similarly, $15^2$ is 15 multiplied by itself, which equals 225. So, we can rewrite the equation as $11y = 324 - 225$. Next, we perform the subtraction: $324 - 225 = 99$. Now our equation looks simpler: $11y = 99$. To isolate $y$, we need to divide both sides of the equation by 11. This gives us $y = \frac99}{11}$. Finally, we perform the division{11} = 9$. Therefore, the value of $y$ is 9. This simple calculation demonstrates the power of algebraic manipulation in solving equations. By following the correct order of operations and isolating the variable, we can efficiently find its value.
Question 58: [Incomplete Question - Needs Clarification]
This question is incomplete as it simply states
