It slopes upward to the right since b 2 is positive

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= 3). It slopes upward to the right (since b = + 2 is positive ), and since the slope is + 2, we know that at any point on the line, an increase (or decrease) in x by 1 unit will be associated with a 2-unit increase (or decrease) in y . M2-2 MATH MODULE 2: LINEAR EQUATIONS

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2. One Point and Slope : If we know the coordinates of one point on the line, say (3, 9), and the slope of the line ( m = 2), then the line in Figure M.2.1 can be defined equally well by the expression m = 2 = ( y – 9)/( x – 3), or ( y – 9) = 2 ( x – 3), (M.2.4) which resolves into y = 3 + 2 x . 3. Two Points : If we have two points on the line, say (–3, –3) and (5, 13), then we can cal- culate the slope m = (13 –[–3])/(5 –[–3]) = 16/8 = 2, and can then use either of the points and the calculated value for the slope to convert the equation into standard form. 4. Implicit Form : Any of the following equations in implicit form can be transformed into standard form: 2 x y = – 3, or 2 x y + 3 = 0, or y – 2 x = 3, or 4 x – 2 y = – 6. (M.2.5) MATH MODULE 2: LINEAR EQUATIONS M2-3 x y 0 (–3, –3) (–1.5, 0) (3, 9) y = 3 + 2 x (5, 13) a = (0, 3) Slope = b = y = + 2 x FIGURE M.2-1
The only situation in which the slope-intercept form will be unsuitable to repre- sent a linear equation occurs with an equation of the form x = a , which is the equa- tion of a vertical line (where the slope is infinite or undefined, depending on which convention you adopt). In this case, if x ≠ 0, the line never cuts the vertical axis, and if x = 0, it coincides with it, so that there is no single intercept. 1.2 INTERPRETING LINEAR EQUATIONS As we have already observed, an equation in the “standard” form, y = a + b x, contains much useful information that is available at a glance. Its vertical intercept is a , its slope is b , and by setting y = 0 and solving for x , we can calculate its horizontal intercept as the point where x = –a/b : ( –a/b , 0). We can do the same with an equation in implicit form: cx + dy = e. Converting the equation to standard form, we have y = e/d – ( c/d ) x. Here, the vertical intercept is e/d , the slope is ( c/d ), and the horizontal intercept is e/c . To take a specific example, if we have the equation 3 x + 5 y = 30, then c = 3, d = 5, and e = 30. Hence the vertical intercept is e/d = 30/ 5 = 6, the slope is ( c/d ) = –3/5, and the horizontal intercept is e/c = 30/3 = 10. The standard form of the equation would be y = e/d – ( c/d ) x = 6 – (3/5) x .

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