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Chapter 4: Problem 6

Verify that the points \((0,2)\) and \((3,0)\) lie on the line $$ 2 x+3 y=6 $$ Hence find the slope of this line. Is the line uphill, downhill or horizontal?

Short Answer

Expert verified
Both points lie on the line. The slope is \(-\frac{2}{3}\), and the line is downhill.

Step by step solution

01

Verify if the point \(0,2\) lies on the line

Substitute \(x = 0\) and \(y = 2\) into the equation \[2x + 3y = 6\].This gives \[2(0) + 3(2) = 6\].Simplify the equation: \[0 + 6 = 6\].Since both sides of the equation are equal, the point \(0, 2\) lies on the line.
02

Verify if the point \(3, 0\) lies on the line

Substitute \(x = 3\) and \(y = 0\) into the equation \[2x + 3y = 6\].This gives \[2(3) + 3(0) = 6\].Simplify the equation: \[6 + 0 = 6\].Since both sides of the equation are equal, the point \(3, 0\) lies on the line.
03

Find the slope of the line

Rewrite the equation in slope-intercept form \[y = mx + b\] by solving for y.Start with \[2x + 3y = 6\].Subtract \[2x\] from both sides: \[3y = 6 - 2x\].Divide both sides by 3: \[y = -\frac{2}{3}x + 2\].The coefficient of \(x\) is the slope \(m = -\frac{2}{3}\).
04

Determine the direction of the line

The slope of the line is \(m = -\frac{2}{3}\).Since the slope is negative, the line is downhill.

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Key Concepts

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

verifying points on a line
To verify if a point lies on a line, you need to see if the coordinates satisfy the line's equation. Let's check if points \( (0,2) \) and \( (3,0) \) are on the line \[ 2x + 3y = 6 \].

For point \( (0,2) \):
  • Substitute \( x = 0 \) and \( y = 2 \) into the equation.
  • This gives \[ 2(0) + 3(2) = 6 \].
  • Since \[ 0 + 6 = 6 \], the equation is true.
Thus, the point \( (0, 2) \) lies on the line.

For point \( (3,0) \):
  • Substitute \( x = 3 \) and \( y = 0 \) into the equation.
  • This gives \[ 2(3) + 3(0) = 6 \].
  • Since \[ 6 + 0 = 6 \], the equation holds true.
Hence, the point \( (3, 0) \) also lies on the line.
slope-intercept form
The slope-intercept form of an equation is a way to express the equation of a line. It's written as \[ y = mx + b \], where \[ m \] is the slope and \[ b \] is the y-intercept.

To convert the equation \[ 2x + 3y = 6 \] into slope-intercept form:
  • Start by isolating \[ y \].
  • Subtract \[ 2x \] from both sides: \[ 3y = 6 - 2x \].
  • Divide by 3: \[ y = - \frac{2}{3}x + 2 \].
Now, the equation is in slope-intercept form \[ y = - \frac{2}{3}x + 2 \], revealing the line's slope and y-intercept.
calculating slope
The slope of a line measures its steepness. Mathematically, slope is the ratio of the change in y to the change in x, often referred to as 'rise over run.'

From the slope-intercept form \[ y = - \frac{2}{3}x + 2 \], we see the slope \[ m = - \frac{2}{3} \]. It's calculated by rearranging and isolating y in the given equation:
  • Start with \[ 2x + 3y = 6 \].
  • Rearrange to \[ 3y = 6 - 2x \].
  • Divide both sides by 3 to get \[ y = - \frac{2}{3}x + 2 \].
The slope, or coefficient of x in this form, is \[ - \frac{2}{3} \]. This indicates that for every 3 units you move horizontally, you move 2 units down.
line direction
The direction of a line depends on its slope. If the slope is positive, the line goes uphill (left to right). If the slope is negative, the line goes downhill.

Our calculated slope is \[ - \frac{2}{3} \]. This means the line is downhill because it decreases in value as you move from left to right.

Key points to remember:
  • Slope > 0: Line goes uphill.
  • Slope < 0: Line goes downhill.
  • Slope = 0: The line is horizontal, flat.
So, with a negative slope, our line is indeed going downhill.

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Most popular questions from this chapter

A monopolist's demand function is given by $$ P+Q=100 $$ Write down expressions for TR and MR in terms of \(Q\) and sketch their graphs. Find the value of \(Q\) which gives a marginal revenue of zero and comment on the significance of this value.

(a) In Section \(4.5\) the following relationship between marginal revenue, \(M R\), and price elasticity of demand, \(E\), was derived: $$ \mathrm{MR}=P\left(1-\frac{1}{E}\right) $$ Use this result to show that at the point of maximum total revenue, \(E=1\). (b) Verify the result of part (a) for the demand function $$ 2 P+3 Q=60 $$

Given the demand function $$ P=1000-2 Q $$ calculate the arc elasticity as \(P\) falls from 210 to 200 .

If the consumption function is $$ C=\frac{300+2 Y^{2}}{1+Y} $$ calculate MPC and MPS when \(Y=36\) and give an interpretation of these results.

If the supply equation is $$ Q=150+5 P+0.1 P^{2} $$ calculate the price elasticity of supply (a) averaged along an arc between \(P=9\) and \(P=11\) (b) at the point \(P=10\)
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