Systems of Equations

Student Summary

A system of equations is a set of 2 or more equations, where the variables represent the same unknown values. For example, suppose that two different kinds of bamboo are planted at the same time. Plant A starts at 6 ft tall and grows at a constant rate of 14\frac14 foot each day. Plant B starts at 3 ft tall and grows at a constant rate of 12\frac12 foot each day. Because Plant B grows faster than Plant A, it will eventually be taller, but when?

We can write equations y=14x+6y = \frac14 x + 6 for Plant A and y=12x+3y = \frac12 x +3 for Plant B, where xx represents the number of days after being planted, and yy represents height. We can write this system of equations.

{y=14x+6 y=12x+3\displaystyle \begin{cases} y = \frac14 x + 6 \\ y = \frac12 x +3 \end{cases}

Solving a system of equations means to find the values of xx and yy that make both equations true at the same time. One way we have seen to find the solution to a system of equations is to graph both lines and find the intersection point. The intersection point represents the pair of xx and yy values that makes both equations true.

Here is a graph for the bamboo example:

&lt;p&gt;Graph of two lines.&lt;/p&gt;<br>  
<p>Graph of two lines, origin O, with grid. Horizontal axis, time in days, scale 0 to 13, by 1’s. Vertical axis, height in feet, scale 0 to 12, by 1’s. A line, labeled Plant A, crosses the y axis at 6. A line, labeled Plant B, crosses the y axis at 3. The lines intersect at the point 12 comma 9.</p>  

The solution to this system of equations is (12,9)(12,9), which means that both bamboo plants will be 9 feet tall after 12 days.

We have seen systems of equations that have no solutions, one solution, and infinitely many solutions.

  • When the lines do not intersect, there is no solution. (Lines that do not intersect are parallel.)
  • When the lines intersect once, there is one solution.
  • When the lines are right on top of each other, there are infinitely many solutions.

Visual / Anchor Chart

Standards

Addressing
8.EE.8

Analyze and solve pairs of simultaneous linear equations. Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs. Solve systems of two linear equations in two variables with integer coefficients: graphically, numerically using a table, and algebraically. Solve real-world and mathematical problems involving systems of two linear equations in two variables with integer coefficients.

8.EE.8.a

Analyze and solve pairs of simultaneous linear equations. Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs. Solve systems of two linear equations in two variables with integer coefficients: graphically, numerically using a table, and algebraically. Solve real-world and mathematical problems involving systems of two linear equations in two variables with integer coefficients.

8.EE.8.b

Analyze and solve pairs of simultaneous linear equations. Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs. Solve systems of two linear equations in two variables with integer coefficients: graphically, numerically using a table, and algebraically. Solve real-world and mathematical problems involving systems of two linear equations in two variables with integer coefficients.