Slope is defined as the steepness of a line and can be calculated using the formula \u201crise over run.\u201d More precisely, this formula is equal to the change in \\(y\\)-value (\u201crise\u201d) divided by the change in \\(x\\)-value (\u201crun\u201d) between two points. To find these changes in values, use \\(y_2-y_1\\) and \\(x_2-x_1\\), where \\(\\left(x_1,y_1\\right)\\) and \\(\\left(x_2,y_2\\right)\\) are points on the line.<\/p>\n
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\\(m=\\frac{\\text{rise}}{\\text{run}}=\\frac{y_2-y_1}{x_2-x_1}\\)<\/p>\n
In this case, let \\(\\left(x_1,\\ y_1\\right)=\\left(0,-3\\right)\\) and \\(\\left(x_2,y_2\\right)=\\left(4,5\\right)\\). Plug each value into the slope formula and simplify.<\/p>\n
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\\(m=\\frac{5-\\left(-3\\right)}{4-0}=\\frac{5+3}{4-0}=\\frac{8}{4}=2\\)<\/p>\n
Therefore, the slope of this line is 2.<\/p>\n<\/div>\n\t\t\t\t\t\tHide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\n\t\t\t\t\t
Question #2:<\/strong>\n\t\t\t\t\t Slope-Intercept and Point-Slope Forms.<\/strong> Cheryl is a real estate agent and has determined that the prices, \\(y\\), of residential properties varies based upon their square footage, \\(x\\), such that each additional square foot adds $100 to the cost of the property. Using information from randomly selected homes in the area, Cheryl determines that the point-slope equation \\(y-205{,}000=100\\left(x-2{,}000\\right)\\) relates price to square footage. What is this equation when converted to slope-intercept form?<\/p>\n<\/div>\n\t\t\t\t\t\\(y=100\\left(x-2{,}000\\right)+205{,}000\\)<\/div>
\\(y=-2{,}000x+205{,}100\\)<\/div>
\\(y=100x+5{,}000\\)<\/div>
\\(y=5{,}000x+100\\)<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t
Show Answer<\/label>\n\t\t\t\t\t\n\t\t\t\t\t\t
Answer:<\/strong>Equations written in slope-intercept form are in the format \\(y=mx+b\\), where \\(m\\) is the slope and \\(b\\) is the \\(y\\)-intercept. In this situation, the cost per square foot is $100. Therefore, the slope \\(m\\) is equal to 100. This can already be seen in the point-slope equation.<\/p>\n
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\\(y-y_1=\\mathbf{m}\\left(x-x_1\\right)\\)<\/p>\n
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\\(y-205{,}000=\\mathbf{100}\\left(x-2{,}000\\right)\\)<\/p>\n
The quickest way to determine \\(b\\)\u2019s value is to rearrange the point-slope equation so that \\(y\\) is by itself on the left and like terms are combined on the right. Add 205,000 to both sides of the equation.<\/p>\n
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\\(y=100\\left(x-2{,}000\\right)+205{,}000\\)<\/p>\n
Now, distribute the 100 to the \\(x\\) and the \u20132,000.<\/p>\n
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\\(y=100x-200{,}000+205{,}000\\)<\/p>\n
Finally, combine like terms to see that \\(b=5{,}000\\).<\/p>\n
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\\(y=100x+5{,}000\\)<\/p>\n
This equation is now in slope-intercept form. From it we can quickly determine that the price of each property is equal to the square-footage times $100, plus a fixed cost of $5,000.<\/p>\n<\/div>\n\t\t\t\t\t\tHide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\n\t\t\t\t\t
Question #3:<\/strong>\n\t\t\t\t\t