# Systems of Equations

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### Systems of Equations

Student Outcomes

• Students solve systems of linear equations in two variables and systems of a linear and a quadratic equation in two variables.
• Students understand that the points at which the two graphs of the equations intersect correspond to the solutions of the system.

### New York State Common Core Math Algebra II, Module 1, Lesson 31

Classwork

Exploratory Challenge 1
a. Sketch the lines given by 𝑥 + 𝑦 = 6 and −3𝑥 + 𝑦 = 2 on the same set of axes to solve the system graphically. Then solve the system of equations algebraically to verify your graphical solution.
b. Suppose the second line is replaced by the line with equation 𝑥 + 𝑦 = 2. Plot the two lines on the same set of axes, and solve the pair of equations algebraically to verify your graphical solution.
c. Suppose the second line is replaced by the line with equation 2𝑥 = 12− 2𝑦. Plot the lines on the same set of axes, and solve the pair of equations algebraically to verify your graphical solution.
d. We have seen that a pair of lines can intersect in 1, 0, or an infinite number of points. Are there any other possibilities?

Exploratory Challenge 2
a. Suppose that instead of equations for a pair of lines, you were given an equation for a circle and an equation for a line. What possibilities are there for the two figures to intersect? Sketch a graph for each possibility.
b. Graph the parabola with equation 𝑦 = 𝑥2. What possibilities are there for a line to intersect the parabola?
Sketch each possibility
c. Sketch the circle given by 𝑥2 + 𝑦2 = 1 and the line given by 𝑦 = 2𝑥 + 2 on the same set of axes. One solution to the pair of equations is easily identifiable from the sketch. What is it?
d. Substitute 𝑦 = 2𝑥 + 2 into the equation 𝑥2 + 𝑦2 = 1, and solve the resulting equation for 𝑥.
e. What does your answer to part (d) tell you about the intersections of the circle and the line from part (c)?

Exercises

1. Draw a graph of the circle with equation 𝑥2 + 𝑦2 = 9.
a. What are the solutions to the system of circle and line when the circle is given by 𝑥2 + 𝑦2 = 9, and the line is given by 𝑦 = 2?
b. What happens when the line is given by 𝑦 = 3?
c. What happens when the line is given by 𝑦 = 4?
2. By solving the equations as a system, find the points common to the line with equation 𝑥 − 𝑦 = 6 and the circle with equation 𝑥2 + 𝑦2 = 26. Graph the line and the circle to show those points.
3. Graph the line given by 5𝑥 + 6𝑦 = 12 and the circle given by 𝑥2 + 𝑦2 = 1. Find all solutions to the system of equations.
4. Graph the line given by 3𝑥 + 4𝑦 = 25 and the circle given by 𝑥2 + 𝑦2 = 25. Find all solutions to the system of equations. Verify your result both algebraically and graphically
5. Graph the line given by 2𝑥 + 𝑦 = 1 and the circle given by 𝑥2 + 𝑦2 = 10. Find all solutions to the system of equations. Verify your result both algebraically and graphically
6. Graph the line given by 𝑥 + 𝑦 = −2 and the quadratic curve given by 𝑦 = 𝑥2 − 4. Find all solutions to the system of equations. Verify your result both algebraically and graphically.

Lesson Summary
Here are some steps to consider when solving systems of equations that represent a line and a quadratic curve.

1. Solve the linear equation for 𝑦 in terms of 𝑥. This is equivalent to rewriting the equation in slope-intercept form. Note that working with the quadratic equation first would likely be more difficult and might cause the loss of a solution.
2. Replace 𝑦 in the quadratic equation with the expression involving 𝑥 from the slope-intercept form of the linear equation. That will yield an equation in one variable.
3. Solve the quadratic equation for 𝑥.
4. Substitute 𝑥 into the linear equation to find the corresponding value of 𝑦.
5. Sketch a graph of the system to check your solution

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