marla and nancy played in a volleyball marathon for charity. they played for 38 h and raised $412....

1.4 - Solving Linear Systems – Substitution

Marla and Nancy played in a volleyball marathon for charity. They played for 38 h and raised $412. Marla was sponsored for $10/h. Nancy was sponsored for $12/h.

How many hours did each student play?Let m represent the number of hours Marla playedLet n represent the number of hours Nancy played1. m + n = 382. 10m + 12n = 412

1. m = 38 - nPlug this into equation 2 and solve for n.

Example #1 Cont’dm = 38 – nPlug this into equation 2: 10m + 12n = 41210(38 – n) + 12n = 412380 – 10n + 12n = 4122n = 32n = 16Plug this back into equation 1. to find m:m = 38 – n = 38 – 16m = 22Therefore, Marla played for 22 hours, and Nancy played for 16 hours.

Challenge Question!Is there any other combination of hours between Marla and Nancy that would add up to 38 hours and $412?

No – two lines only intersect once, so this

solution is the only one.

Example #2Most gold jewellery is actually a mixture of gold and copper. A jeweller is reworking a few pieces of old gold jewellery into a new necklace. Some of the jewellery is 84% gold by mass, and the rest is 75% gold by mass. The jeweller needs 15.00 g of 80% gold for the necklace. How much of each alloy should he use?Let x represent the mass of the 84% alloy in grams.Let y represent the mass of the 75% alloy in grams.

Make 2 equations, substitute one equation into the other to find the point of intersection.

Example #2 Cont’dLet x represent the mass of the 84% alloy in grams.Let y represent the mass of the 75% alloy in grams.

1. x + y = 152. 0.84x + 0.75y = 12 (12 = 15 x 0.80)

1. y = 15 – x2. 0.84x + 0.75(15 – x) = 12

0.84x + 11.25 – 0.75x = 120.09x = 0.75x =

x = 8.33

y = 15 – x = 15 – 8.33y = 6.67

Therefore, in order to make 15.00 g of an 80.0 % alloy, the jeweller needs to mix 8.33 g of the 84% alloy, and 6.67 g of the 75% alloy.

Example #3Determine, without graphing, where the lines with equations 5x + 2y = -2 and 2x – 3y = -16 intersect.

1. 5x + 2y = -22. 2x – 3y = -16Rearrange “1.” for y and substitute into equation “2.”:1. 2y = -2 – 5x

y =

2. 2x – 3() = -162x + 3 + = -16multiply everything by 2:4x + 6 + 15x = -3219x = -38 x = -2

y =

=

=

y = 4Therefore, the lines intersect at (-2, 4)

In Summary…• To determine the solution to a system of linear

equations algebraically:• Isolate one variable in one of the equations• Substitute the expression for this variable into

the other equation• Solve the resulting linear equation• Substitute the solved value into one of the

equations to determine the value of the other variable