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 $x$ represents yards of silver thread and $y$ yards of gold thread. 

• The length constraint: $x + y = 240$
• The cost constraint: $0.04x + 0.07y = 15$

Every point on the graph of $x+y=240$ is a pair of values that meets the length constraint.

Every point on the graph of $0.04x + 0.07y = 15$ is a pair of values that meets the cost constraint.

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.  

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That point is $(60, 180)$, which represents 60 yards of silver thread and 180 yards of gold thread.

If we substitute 60 for $x$ and 180 for $y$ in each equation, we find that these values make the equation true. $(60,180)$ is a solution to both equations simultaneously. 

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.

$\begin {cases} x + y = 240\\0.04x + 0.07y = 15 \end {cases}$

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.