In this Grade 9 lesson from California Reveal Math Algebra 1, students learn to solve systems of linear and quadratic equations using both graphical and algebraic methods, including substitution and factoring with the Zero Product Property. Students explore how the relative positions of a parabola and a line determine whether a system has zero, one, or two points of intersection. The lesson also applies these skills to real-world contexts, such as finding when two revenue models represented by quadratic and linear functions produce equal values.
Section 1
Solve Linear-Quadratic Systems by Graphing
To solve a linear-quadratic system by graphing:
Step 1. Identify that one equation is linear and one is quadratic.
Step 2. Graph the linear equation (line).
Step 3. Graph the quadratic equation (parabola) on the same coordinate system.
Step 4. Determine whether the graphs intersect.
Step 5. Identify the points of intersection.
Step 6. Check that each ordered pair satisfies both original equations.
Section 2
Solve Linear-Quadratic Systems by Substitution
To solve a linear-quadratic system by substitution:
Step 1. Identify which equation is linear and which is quadratic.
Step 2. Solve the linear equation for either variable.
Step 3. Substitute the expression from Step 2 into the quadratic equation.
Step 4. Solve the resulting quadratic equation.
Step 5. Substitute each solution from Step 4 into the linear equation to find the other variable.
Step 6. Write each solution as an ordered pair and check it in both original equations.
Section 3
Number of Solutions in a Linear-Quadratic System
A system of one linear equation and one quadratic equation can have 0, 1, or 2 real solutions, depending on how many times the line intersects the parabola.
To determine the number of solutions algebraically, substitute the linear expression into the quadratic equation to obtain a single quadratic equation of the form:
Expand to review the lesson summary and core properties.
Section 1
Solve Linear-Quadratic Systems by Graphing
To solve a linear-quadratic system by graphing:
Step 1. Identify that one equation is linear and one is quadratic.
Step 2. Graph the linear equation (line).
Step 3. Graph the quadratic equation (parabola) on the same coordinate system.
Step 4. Determine whether the graphs intersect.
Step 5. Identify the points of intersection.
Step 6. Check that each ordered pair satisfies both original equations.
Section 2
Solve Linear-Quadratic Systems by Substitution
To solve a linear-quadratic system by substitution:
Step 1. Identify which equation is linear and which is quadratic.
Step 2. Solve the linear equation for either variable.
Step 3. Substitute the expression from Step 2 into the quadratic equation.
Step 4. Solve the resulting quadratic equation.
Step 5. Substitute each solution from Step 4 into the linear equation to find the other variable.
Step 6. Write each solution as an ordered pair and check it in both original equations.
Section 3
Number of Solutions in a Linear-Quadratic System
A system of one linear equation and one quadratic equation can have 0, 1, or 2 real solutions, depending on how many times the line intersects the parabola.
To determine the number of solutions algebraically, substitute the linear expression into the quadratic equation to obtain a single quadratic equation of the form: