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Chapter 3: Problem 1

Find the slope and one point on each line. $$y-5=2(x-3)$$

Short Answer

Expert verified
The slope of the line is \(2\) and one point on the line is \((3,5)\).

Step by step solution

01

Finding the slope

The given equation, \(y - 5 = 2(x - 3)\), is already in point-slope form, so the number in front of the \(x\) is the slope of the line. Therefore, the slope \(m\) is \(2\).
02

Identifying a point on the line

In the equation \(y - 5 = 2(x - 3)\), the number next to the \(y\) on the left side and the number inside the bracket on the right side correspond to the \(y\) and \(x\) coordinates of a point on the line, respectively. This means that the point \((3,5)\) is on the line.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Slope
The concept of the slope is crucial in understanding how lines behave on a graph. In algebra, the slope of a line is a number that describes both its direction and steepness. You can think of it as a measure of how much the line "rises" vertically compared to how much it "runs" horizontally. This is mathematically represented by the formula:\[m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}\]where
  • \(m\) is the slope.
  • \(y_2 - y_1\) is the change in the \(y\)-coordinate (rise).
  • \(x_2 - x_1\) is the change in the \(x\)-coordinate (run).
In the point-slope form of a linear equation, the slope is the coefficient of the \(x\)-term. In our exercise, the equation \(y - 5 = 2(x - 3)\) shows that the slope is \(2\), indicating a line that rises two units up for every unit it moves to the right.
Point-Slope Form
The point-slope form is one of the various ways to represent a linear equation. It's particularly helpful when you know one point on the line and the slope. The general formula for the point-slope form is: \[y - y_1 = m(x - x_1)\]where:
  • \((x_1, y_1)\) is a point on the line.
  • \(m\) is the slope.
This form is useful for quickly identifying both the slope and a specific point on a line. In our exercise, the equation is \(y - 5 = 2(x - 3)\). Here, \(m = 2\) and the point \((x_1, y_1)\) is \((3, 5)\). By utilizing point-slope form, we can easily draft a rough idea of what the graph of the line would look like.
Linear Equations
Linear equations are equations that represent straight lines when graphed on the Cartesian plane. They are foundational in algebra and are described in several forms, such as the slope-intercept form, point-slope form, and standard form. In any linear equation, the highest power of the variable, typically \(x\), is 1. This ensures the equation forms a straight line when plotted.Key points about linear equations include:
  • They have constant slopes, meaning the steepness doesn鈥檛 change.
  • Each equation can be converted to different forms depending on available information (e.g., slope-intercept or standard form).
  • The graph is always a straight line.
In our exercise, the linear equation \(y - 5 = 2(x - 3)\) given in point-slope form can be manipulated into other forms if needed, thus offering flexibility in understanding the line's attributes and how it behaves across the graph.

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Most popular questions from this chapter

Give examples of an equation in one variable and an equation in two variables. How do their solutions differ?

Express each variation as an equation. Then find the requested value. Assume that all variables represent positive numbers. \(F\) varies directly with the product of \(m_{1}\) and \(m_{2}\) and inversely with \(d^{2} .\) If \(F=1,250\) when \(m_{1}=400\) and \(m_{2}=500\) and \(d=100,\) find \(m_{1}\) when \(F=1,550\) and all other values remain constant.

Express each joint variation as an equation. Then find the requested value. Assume that all variables represent positive numbers. SEE EXAMPLE 3. (OBJECTIVE 3) \(z\) varies jointly with \(r\) and the square of \(s .\) If \(z=24\) when \(r\) and \(s\) are 2 , find \(z\) when \(r=3\) and \(s=4\).

Express each variation as an equation. Then find the requested value. Assume that all variables represent positive numbers. \(y\) varies inversely with \(x^{2} .\) If \(y=6\) when \(x=4,\) find \(y\) when \(x=2\).

Express each direct variation as an equation. Then find the requested value. Assume that all variables represent positive numbers. SEE EXAMPLE 1. (OBJECTIVE 1) \(r\) varies directly with \(s .\) If \(r=21\) when \(s=6,\) find \(r\) when \(s=12\).
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