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Chapter 1: Problem 46

Find an equation of the line that passes through the point \((-1,3)\) and is parallel to the line passing through the points \((-2,-3)\) and \((2,5)\)

Short Answer

Expert verified
The equation of the line that passes through the point \((-1,3)\) and is parallel to the line passing through the points \((-2,-3)\) and \((2,5)\) is: \(y = 2x + 5\).

Step by step solution

01

Find the slope of the line passing through the points \((-2,-3)\) and \((2,5)\)

The slope of a line passing through the points \((x_1, y_1)\) and \((x_2, y_2)\) can be found using the formula: \[m = \frac{y_2 - y_1}{x_2 - x_1}\] Let \((-2,-3)\) be \((x_1, y_1)\) and \((2,5)\) be \((x_2, y_2)\). Plugging these values into the formula, we get: \[m = \frac{5 - (-3)}{2 - (-2)}\]
02

Simplify the expression for the slope

Now we simplify the expression obtained in Step 1: \[m = \frac{5 + 3}{2 + 2}\] \[m = \frac{8}{4}\] \[m = 2\] So, the slope of the line passing through the given points is 2.
03

Use the point-slope form of the equation

Since the given line and the required line are parallel, they will have the same slope. Therefore, the slope of the required line is also 2. Now we will use the point-slope form of the equation, which is given by: \[y - y_1 = m(x - x_1)\] Since the required line passes through the point \((-1, 3)\), we can plug in \((-1, 3)\) as \((x_1, y_1)\), and use the slope of 2: \[y - 3 = 2(x - (-1))\]
04

Simplify the equation to get the final answer

Simplify the equation obtained in Step 3: \[y - 3 = 2(x + 1)\] Distribute the 2 to both terms in the parentheses: \[y - 3 = 2x + 2\] Add 3 to both sides of the equation to isolate y: \[y = 2x + 5\] So, the equation of the line that passes through the point \((-1,3)\) and is parallel to the line passing through the points \((-2,-3)\) and \((2,5)\) is: \(y = 2x + 5\).

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Most popular questions from this chapter

The relationship between temperature measured in the Celsius scale and the Fahrenheit scale is linear. The freezing point is \(0^{\circ} \mathrm{C}\) and \(32^{\circ} \mathrm{F}\), and the boiling point is \(100^{\circ} \mathrm{C}\) and \(212^{\circ} \mathrm{F}\). a. Find an equation giving the relationship between the temperature \(F\) measured in the Fahrenheit scale and the temperature \(C\) measured in the Celsius scale. b. Find \(F\) as a function of \(C\) and use this formula to determine the temperature in Fahrenheit corresponding to a temperature of \(20{ }^{\circ} \mathrm{C}\). c. Find \(C\) as a function of \(F\) and use this formula to determine the temperature in Celsius corresponding to a temperature of \(70^{\circ} \mathrm{F}\).

Find an equation of the line that passes through the point \((-2,2)\) and is parallel to the line \(2 x-4 y-8=0\).

For each pair of supply-and-demand equations, where \(x\) represents the quantity demanded in units of 1000 and \(p\) is the unit price in dollars, find the equilibrium quantity and the equilibrium price. $$ p=-0.3 x+6 \text { and } p=0.15 x+1.5 $$

Suppose the demand-and-supply equations for a certain commodity are given by \(p=a x+b\) and \(p=c x+d\), respectively, where \(a<0, c>0\), and \(b>d>0\) (see the accompanying figure). a. Find the equilibrium quantity and equilibrium price in terms of \(a, b, c\), and \(d\). b. Use part (a) to determine what happens to the market equilibrium if \(c\) is increased while \(a, b\), and \(d\) remain fixed. Interpret your answer in economic terms. \(\mathbf{c}\). Use part (a) to determine what happens to the market equilibrium if \(b\) is decreased while \(a, c\), and \(d\) remain fixed. Interpret your answer in economic terms.

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If \(p=m x+b\) is a linear demand curve, then it is generally true that \(m<0\).
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