Read the following problem: Your bank account increases linearly each week. If after 20 weeks of work, your bank account is at $560, while after 21 weeks of work it is at $585, find a way to express the relationship between how much money you’ve earned and how many weeks you’ve worked in slope-intercept form.
Notice that the problem states, “Your bank account increases linearly each week,” meaning that you are saving the same amount of money each time, which means it will have a smooth slope. That “smooth,” uniformly consistent savings plan makes it linear. If you don’t save the same amount all the time, then it is not linear.
If you started with $560 and now have $585 the next week, then you have earned $25 after 1 week of work. You can figure this out by subtracting $560 from $585. $585−$560=$25{\displaystyle $585-$560=$25}.
If you had $560 after 20 weeks of work, and you know that you earn $25 after every week of work, then you can multiply 20 \times 25 to figure out how much money you earned in those 20 weeks. 20×25=500{\displaystyle 20\times 25=500}, which means that you earned $500 in those weeks. Since you have $560 after 20 weeks and have earned $500, you can figure out how much you started with by subtracting 500 from 560. 560 - 500 = 60. Therefore, b=60{\displaystyle b=60}.
y=mx+b{\displaystyle y=mx+b} Substitute m{\displaystyle m} (slope) and b{\displaystyle b} (y-intercept) as follows: y=25x+60{\displaystyle y=25x+60}
How much money have you earned after 10 weeks? Substitute x{\displaystyle x} with 10{\displaystyle 10} in this equation to find out: y=25x+60{\displaystyle y=25x+60} y=25(10)+60{\displaystyle y=25(10)+60} y=250+60{\displaystyle y=250+60} y=310{\displaystyle y=310}. After 10 weeks, you’ve made $310. Notice how y{\displaystyle y} is the (manipulated/dependent variable). How many weeks would you have to work to earn 800 dollars? Plug “800” into the y{\displaystyle y} variable of the equation to get the x{\displaystyle x} value. y=25x+60{\displaystyle y=25x+60} 800=25x+60{\displaystyle 800=25x+60} 800−60{\displaystyle 800-60} 25x=740{\displaystyle 25x=740} 25x25=74025{\displaystyle {\frac {25x}{25}}={\frac {740}{25}}} x=29. 6{\displaystyle x=29. 6}. You can earn 800 dollars in almost 30 weeks.
4y + 3x = 16 = 4y + 3x - 3x = -3x +16 (by subtraction) 4y = -3x +16 (by rewriting, simplifying the subtraction)
4y = -3x +16 = 4/4y = -3/4x +16/4 = (by division) y = -3/4x + 4 (by rewriting, simplifying the division)
y = -6, m = 4, x = -1 (the given values) y = mx + b (the formula) -6 = (4)(-1) + b (by substitution)
-6 = (4)(-1) + b -6 = -4 + b (by multiplying) -6 - (-4) = -4 -(-4) + b (by subtraction) -6 - (-4) = b (simplifying the right hand side) -2 = b (simplifying the left hand side)
m = 4, b = -2 y = mx + b y = 4x -2 (by substitution)
(Y2 - Y1) / (X2 - X1) = (2 - 4)/(1 - -2) = -2/3 = m The slope of the line is -2/3.
y = 2, x, = 1, m = -2/3 y = mx + b 2 = (-2/3)(1) + b 2 = -2/3 + b 2 - (-2/3) = b 2 + 2/3 = b, or b = 8/3
y = mx + b y = -2/3x + 2 2/3
If your slope is negative, then you’ll either have to move the y-coordinate up instead of down, or move the x-coordinate to the left instead of the right. You’ll get the same result either way.
You are working where m = -2 as the slope of a line and the coordinates of a point are (4, -3), and these are our (x1,y1) like any defined point on the line. So, using those given values we have: y - y1 = m(x - x1), y - (-3) = -2(x - 4), by substitution using the point and slope y + 3 = -2(x - 4), by simplifying -(-3) to + 3 y + 3 = -2x + -2(-4), by distribution y + 3 = -2x + 8, by multiplying y + 3 - 3 = -2x + 8 - 3, by subtraction (of equals from both sides of the equation)y = -2x + 5, by simplifying/rewriting it (That fits the y = mx + b called the Slope Intercept Form). What is Point–-slope form based on? The point-slope form expresses the fact that the difference of y values for two points on one line (that is, y − y1) can be stated as directly proportional to the difference of x values (that is, x − x1). There is a proportionality constant called m (the slope of the line). We find that Direct Proportion is a comparison that can be stated in a form similar to y = kx. Here we notice that y - y1 = m(x - x1) fits the form y = kx . Direct proportion means that given two variables such as x and y, then y is called directly proportional to x, if there is a constant k such that y = kx, if and only if x is not zero. “k” is the proportionality constant which is just the slope as we are using it. (You can also express direct proportion by saying “x and y vary directly”, or express that “x and y are in direct variation”).
