A linear-quadratic system is a set of equations where at least one is linear and at least one is quadratic. They are expressed identical to linear systems. Similar to systems of linear equations, linear-quadratic systems contain more than one unknown variable, and the solution is the set of coordinates that make all equations true simultaneously. In the example above, the solution is and These coordinates make the sides equal in both equations. The solution is usually written as a point:They can be solved graphically, by substitution or by elimination.
Similar to solving linear systems, a linear-quadratic system can be solved by graphing both functions in the system. The solution(s) can be found by identifying the coordinates of the point(s) of intersection.
Solve the linear-quadratic system graphically.
Solving a linear-quadratic system graphically means finding the point or points where the line and the parabola intersect. Since the linear equation is given in slope-intercept form, we can see that the -intercept of the line is and the slope is Since the quadratic function is given in standard form, we can see the following characteristics of the parabola. We'll use the characteristics to graph both functions on the same coordinate plane.
From the graph we can see that the parabola and the line do not intersect. Thus, there are no solutions to the system.
First, it is necessary to substitute one equation into the other. It does not matter which equation is chosen. Here, will be substituted into
The -coordinates of the points of intersections are and They can be used to find the corresponding -values. To do this, substitute them into either equation and solve. Here, will be used. Thus, the solutions to the system are
The equations can now be combined, by either adding or subtracting one of the equations from the other, whichever will lead to eliminating one of the variables. In this example, the coefficients of have the same sign. Therefore, subtracting one equation from the other will eliminate The second equation will now be subtracted from the first, though it is an arbitrary choice. This is done by subtracting the left-hand side of the second equation from the first, and doing the same for the right-hand side.