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Chapter 4: Problem 28

The line \(3 y=7 x-2\) has an intercept of \(-2\) on the \(y\)-axis.

Short Answer

Expert verified
The correct intercept is \(-\frac{2}{3}\), not -2.

Step by step solution

01

Identify the line equation

Recognize the given line equation: \[ 3y = 7x - 2 \]
02

Express the equation in slope-intercept form

The slope-intercept form of a line is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. To convert the given equation into this form, divide both sides by 3: \[ y = \frac{7}{3}x - \frac{2}{3} \]
03

Identify the y-intercept

From the equation in slope-intercept form, identify the intercept term, which is the constant term \( b \): \[ b = -\frac{2}{3} \]
04

Compare with given intercept

Compare the identified y-intercept with the given intercept of -2. Notice that the actual intercept from the equation is \(-\frac{2}{3}\), not -2.

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Key Concepts

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

Linear Equations
A linear equation is an equation that represents a straight line on a graph. The general form of a linear equation is \(Ax + By = C\), where \(A\), \(B\), and \(C\) are constants. Linear equations have no exponents higher than 1 and only involve two variables, typically x and y.
To graph a linear equation, you need to find two points that satisfy the equation and draw a line through them. However, there is a much simpler way: by converting the equation into its slope-intercept form, you can easily identify the slope and y-intercept, which helps you plot the line quickly.
Y-Intercept
The y-intercept is the point where the line crosses the y-axis. In the slope-intercept form of a linear equation, \(y = mx + b\), the y-intercept is represented by the constant \(b\).
To find the y-intercept from an equation not in slope-intercept form, you first need to transform it into \(y = mx + b\). Let's use our exercise's example equation:\(3y = 7x - 2\).
  • Divide both sides by 3 to isolate y: \(y = \frac{7}{3}x - \frac{2}{3}\).

Here, \(-\frac{2}{3}\) is our y-intercept, as it is the value of y when x is 0. Often, students get confused about identifying the y-intercept, but remember that in \(y = mx + b\), \(b\) is always the y-intercept.
Slope
The slope is a measure of how steep a line is. In the equation \(y = mx + b\), the slope is represented by \(m\). The slope indicates how much y changes for a change in x.
For our example, the equation \(3y = 7x - 2\) becomes \(y = \frac{7}{3}x - \frac{2}{3}\) when converted into slope-intercept form. Here, the slope is \(\frac{7}{3}\).
This means that for every unit increase in x, y increases by \(\frac{7}{3}\) units. The slope is important because it tells you the direction and steepness of the line. A positive slope means the line goes upwards, and a negative slope means the line goes downwards. Slope is fundamental for understanding how linear equations behave on a graph.

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Most popular questions from this chapter

The curves \(y=x^{2}, \quad y=x(2-x)\) intersect at: (a) \((0,0),(1,1)\) (b) \((2,4)\) (c) \((0,0),(2,4)\) (d) \((0,0),(-1,1)\) (e) \((1,1)\).

Find the equation of the line through \(\mathrm{A}(5,2)\) which is perpendicular to the the line \(y=3 x-5\). Hence find the coordinates of the foot of the perpendicular from A to the line.

Write down the equation of the line which goes through the point \((7,3)\) and which is inclined at \(45^{\circ}\) to the positive direction of the \(x\)-axis. Find the area enclosed by this line and the coordinate axes.

\(\mathrm{A}, \mathrm{B}\) and \(\mathrm{C}\) are three points where: (a) \(\mathrm{C}\) is equidistant from \(\mathrm{A}\) and \(\mathrm{B}\). (b) The coordinates of \(\mathrm{A}, \mathrm{B}\) and \(\mathrm{C}\) are \((1,2),(3,6)\) and \((2,4)\).

Show that the triangle whose vertices are \((1,1),(3,2),(2,-1)\) is isosceles.
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