Unit 4, Lesson 13
Solving Systems of Equations
Grade 8
13.1
Warm-up
<p>Graph of 2 lines, origin O. Horizontal axis, scale negative 25 to 25, by 5’s. Vertical axis, scale negative 20 to 20, by 5’s. A line is labeled y equals negative x plus 10. Another line is labeled y equals 2 x plus 4. The lines intersect at the point 2 comma 8.</p>
13.2
Activity
Here are three systems of equations. Find the solution to each system.
Match each graph to one of the systems of equations, then use the graphs to check that your solutions are reasonable.
<p>A graph with two lines, in the x y plane, origin O. The graph has a scale of negative 25 to 25 on both the x and y axis. One line slants downward and right. It crosses the x axis at 10 and the y axis at 20. Another line slants upward and right. It crosses the y axis at 5. It crosses the x axis to the left of the origin. </p>
<p>A graph with two lines, in the x y plane, origin O. The graph has a scale of negative 25 to 25 on both the x and y axis. One line slants upward and right. It crosses the y axis between 25 and 30. It crosses the x axis between negative 10 and negative 15. Another line slants upward and right. It crosses the y axis between 10 and 15. It crosses the x axis between negative 20 and negative 25. </p>
<p>A graph with two lines, in the x y plane, origin O. The graph has a scale of negative 25 to 25 on both the x and y axis. One line slants upward and right. It crosses the x axis between 0 and 5. It crosses the y axis between 0 and negative 5. Another line slants upward and right. It crosses the x axis at 5 and the y axis at negative 10. </p>
Student Lesson Summary
Sometimes it is easier to solve a system of equations without having to graph the equations and look for an intersection point. In general, whenever we are solving a system of equations written as
we know that we are looking for a pair of values that makes both equations true. In particular, we know that the value for will be the same in both equations. That means that
For example, look at this system of equations:
Since the value of the solution is the same in both equations, then we know that:
We can solve this equation for :
But this is only half of what we are looking for: we know the value for , but we need the corresponding value for .
Since both equations have the same value, we can use either equation to find the -value: or .
In both cases, we find that . So the solution to the system is . We can verify this by graphing both equations in the coordinate plane.
<p>Graph of two lines line, origin O, with grid. Horizontal axis, x, scale negative 4 to 1, by 1s. Vertical axis, y, scale negative 1 to 4, by 1’s. The lines intersect at the point negative 2 comma 2. </p>
In general, a system of linear equations can have:
- No solutions. In this case, the lines that correspond to each equation never intersect. They have the same slope and different -intercepts.
- Exactly one solution. The lines that correspond to each equation intersect in exactly one point. They have different slopes.
- An infinite number of solutions. The graphs of the two equations are the same line! They have the same slope and the same -intercept.
None