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Chapter 1: Problem 17
If the line passing through the points \((1, a)\) and \((4,-2)\) is parallel to the line passing through the points \((2,8)\) and \((-7, a+4)\), what is the value of \(a\) ?
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Determine whether the points lie on a straight line. $$ A(-2,1), B(1,7), \text { and } C(4,13) $$
Show that two distinct lines with equations \(a_{1} x+b_{1} y+\) \(c_{1}=0\) and \(a_{2} x+b_{2} y+c_{2}=0\), respectively, are parallel if and only if \(a_{1} b_{2}-b_{1} a_{2}=0\). Hint: Write each equation in the slope-intercept form and compare.
Find an equation of the line that satisfies the given condition. \text { The line passing through }(-3,4) \text { and parallel to the } x \text { -axis }
Let \(L_{1}\) and \(L_{2}\) be two nonvertical straight lines in the plane with equations \(y=m_{1} x+b_{1}\) and \(y=m_{2} x+b_{2}\), respectively. Find conditions on \(m_{1}, m_{2}, b_{1}\), and \(b_{2}\) such that (a) \(L_{1}\) and \(L_{2}\) do not intersect, (b) \(L_{1}\) and \(L_{2}\) intersect at one and only one point, and (c) \(L_{1}\) and \(L_{2}\) intersect at infinitely many points.
For each supply equation, where \(x\) is the quantity supplied in units of 1000 and \(p\) is the unit price in dollars, (a) sketch the supply curve and (b) determine the number of units of the commodity the supplier will make available in the market at the given unit price. $$ p=2 x+10 ; p=14 $$
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