Linear System Skills

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Algebra

Skills:

Slope and y-intercept form: y = mx +b

Standard form: ax +by +c = 0

Rules:

1.

X must lead and be positive

2.

No fraction

Graphing Strategies using standard forms (the x and y intercepts):

ax +by =C

Y-intercept is

when x is zero

X-intercept is

when y is zero

Find the two

intercepts on the graph, join them.

1.

Don’t use this strategy if the intercepts are going to be fractions,

because it won’t be accurate, use y=mx+b

Writing Equations of Lines (using delta y delta x strategy)

2.

Find slope first

3. m = ∆y/∆x (by substituting one pair of points)

Solving Equations with Rational Coefficients

Use scalar

Solve Linear System (using the method of substitution 1)

y = 3x −4 (expression for

y),

Substitute into x +y = 8

x+3x-4=8

Solve Linear System (using the method of substitution 2)

y1 = ax+b

y2 = cx+d

y1=y2

ax+b = cx+d

Solve Linear System (using the method of elimination)

ax+y=c

dx+y=e (subtraction)

gx+y =o (y is eliminated to get x)

Tip: addition is always better than subtraction)

Real Life Word problems

A vending machine contains a total of 395 quarters and dimes. The total value of the coins is $66.80. How many of each coin are required to make this amount?

q + d = 395 0.25q + 0.10d = 66.80

A chemistry student was asked to make 25mL of 48% alcohol by volume by mixing 40% alcohol solution by volume and 60% alcohol solution by volume. How many litres of each alcohol solution must the student use to make 25mL of 48% alcohol?

a40 + a60 = 25 0.40a40 + 0.60a60 = 0.48(25)

A store owner sells peanuts of $4.80/kg and raisins for $2.40/kg. She will make a mixture of peanuts and raisins called Nutsins and sell it for $3.36/kg. How many kilograms of peanuts and raisins does she need to make 200kg of Nutsins?

r+ p = 200 4.80p + 2.40r = 3.36(200)

The length of a rectangle is 19 m longer than its width. If the perimeter is 606 m, what are the dimensions of the rectangle?

l = w + 19 2l + 2w = 606

When 4 times the larger of two numbers is added to 3 times the smaller the result is 68. Seven times the larger number less 5 times the smaller number is 37. What are the numbers?

4l + 3s = 68 7l −5s = 37

Jeanne took a trip from Brownsville to Montreal, a distance of 830km. She travelled part of the way by bus and the rest of the way by plane. The bus averaged 40k/h and the plane averaged 500k/h. The whole trip took 3.5 hours. How many hours were spent on the bus?

p+b = 3.5 500p + 40b = 830

A tailor repairs cuffs on shirts and hems pants. He can repair 2 cuffs per hour and hem 10 pants in one hour. In an 8 hour day, how long did he spend repairing cuffs and how long did he hem pants if 56 articles of clothing were worked on?

c +ℎ = 8 2c +10ℎ = 56

A chocolate chip recipe requires 13kg of butter and 8 kg of oatmeal. A low fat oatmeal cookie recipe calls for 2kg of butter and 29 kg of oatmeal. You have 47kg of butter and 140 kg of oatmeal in the pantry. How many batches of chocolate chip cookies and how many batches of oatmeal cookies can you make?

13c + 2a= 47 8c +29a = 140

Geometry property

________

length of a line: |ab|=/∆y2+∆X2

midpoints of a line: (x1+x2 / 2), (y1+y2 / 2)

paralell lines: m1=m2

perpendicular lines(right angles): m1=-1/m2

opposite side equal length? |ab|=|dc|

does the two lines bisect?

1.find the midpoint

2.find the point of intersection (slope first, delta x and delta y)

3.see if the two are the same

Distance between a point to a line:

always perpendicular, because it has the shortest distance

1. get equation for the line, negative reciprocal=slope, delta x and delta y

2. solve system using the two equations of the two lines

3. find the distance using the point and the point of intersection, |ab|=/∆y2+∆X2

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