Problem 59 - BIG IDEAS MATH Algebra 2: Common Core Student Edition 2015

Question

Solve the system by graphing. \(2 x^2-3 x-y=0\) \(\frac{5}{2} x-y=\frac{9}{4}\)

Step-by-Step Solution

Answer

The solution will depend on the point of intersection of the two graphs

Step 1: Graph the First equation

Plot the graph of the first equation \(2 x^2-3 x-y=0\). This is a parabolic curve.

Step 2: Graph the Second equation

Plot the graph of the second equation \(\frac{5}{2} x-y=\frac{9}{4}\). This will be a straight line.

Step 3: Find the Point of Intersection

The solution to the system is the point(s) where these two graphs intersect. By visually inspecting the graphs, or by solving the system algebraically, we can find the point of intersection, which is the solution of the system.

Key Concepts

Graphing Quadratic Equations

Solving systems of equations often requires graphing quadratic equations. A quadratic equation is typically in the form of \( ax^2 + bx + c = 0 \), and its graph is a curve called a parabola. While graphing, the first step is to identify the vertex, which is the highest or lowest point of the parabola. Then, find the axis of symmetry, which is a vertical line that divides the parabola into two symmetrical halves.

To graph a quadratic equation, you can also find additional points by choosing values for \( x \) and solving for \( y \) to get coordinate pairs. Plot these on the coordinate plane and draw a smooth curve through them. The direction in which the parabola opens (upward or downward) depends on the sign of \( a \). If \( a > 0 \) the parabola opens upwards, if \( a < 0 \) it opens downwards.

Graphing Linear Equations

Graphing linear equations is another fundamental concept when solving systems by graphing. A linear equation can be written in the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. The slope indicates the steepness and the direction of the line, while the y-intercept is the point where the line crosses the y-axis.

To graph a linear equation, start by plotting the y-intercept on the y-axis. Then, use the slope to find another point. For example, a slope of \( \frac{5}{2} \) means you can go up 5 units and right 2 units from the y-intercept to find the second point. Connect these two points with a straight line extending in both directions. This line represents all the solutions to the equation.

Points of Intersection

The points of intersection are the coordinates where the graphs of two or more equations meet on the coordinate plane. When solving systems by graphing, we’re interested in finding all such points, as they represent the solutions to the system of equations. In the case of a linear and a quadratic equation, there can be zero, one, or two points of intersection.

To find these points graphically, plot both equations on the same set of axes and look for where the graphs cross. Algebraically, you would set the equations equal to each other and solve for \( x \). Once you find \( x \), substitute it back into either equation to solve for \( y \). The resulting \( (x, y) \) coordinate pairs are your solutions. In systems with no solutions the graphs do not intersect, and with an infinite number of solutions, the graphs coincide.