Unit 4, Lesson 12
Writing and Graphing Systems of Linear Equations
Integrated Math 1
12.1
Warm-up
Math Talk: A Possible Mix?
Diego bought some raisins and walnuts to make trail mix.
Raisins cost $4 a pound and walnuts cost $8 a pound. Diego spent $15 on both ingredients.
Decide if each pair of values could be a combination of raisins and walnuts that Diego bought.
- 4 pounds of raisins and 2 pounds of walnuts
- 1 pound of raisins and 1.5 pounds of walnuts
- 2.25 pounds of raisins and 0.75 pounds of walnuts
- 3.5 pounds of raisins and 1 pound of walnuts
Student Lesson Summary
A costume designer needs some silver and gold thread for the costumes for a school play. She needs a total of 240 yards. At a store that sells thread by the yard, silver thread costs $0.04 a yard and gold thread costs $0.07 a yard. The designer has $15 to spend on the thread.
How many of each color should she get if she is buying exactly what is needed and spending all of her budget?
This situation involves two quantities and two constraints—length and cost. Answering the question means finding a pair of values that meets both constraints simultaneously. To do so, we can write two equations and graph them on the same coordinate plane.
Let
- The length constraint:
- The cost constraint:
Every point on the graph of
Every point on the graph of
The point where the two graphs intersect gives the pair of values that meets both constraints.
Graph of 2 intersecting lines, origin O. Horizontal axis 0 to 280, by 20’s, labeled, yards of silver thread. Vertical from 0 to 280, by 20’s, labeled yards of gold thread. Line 1 starts at 240 comma 0, passes through 60 comma 180, ends at 240 comma 0. Line 2 starts at 0 comma 214 point 2 9, passes through 60 comma 180, ends at 280 comma 54 point 2 9.
That point is
If we substitute 60 for
Two or more equations that represent the constraints in the same situation form a system of equations. A curly bracket is often used to indicate a system.
The solution to a system of equations is a pair of values that makes all of the equations in the system true. Graphing the equations is one way to find the solution to a system of equations.
solution to a system of equations
A solution to a system of equations is the values for the variables that make all the equations true.
This graph shows a system of two equations. The solution of the system is a coordinate pair that makes both equations true. On the graph, the solution is shown as the point where the two lines intersect.
system of equations
Two or more equations that represent the constraints in the same situation form a system of equations.
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