Problem 70 - Precalculus with Limits

Question

In Exercises 57-70, find any points of intersection of the graphs algebraically and then verify using a graphing utility. \(5x^2-2xy+5y^2-12=0\) \(x+y-1=0\)

Step-by-Step Solution

Answer

The points of intersection of the two given equations are \((\frac{3}{2},-\frac{1}{2})\) and \((\frac{-4}{3},\frac{7}{3})\)

Step 1: Simplify the Second Equation

From the equation \(x+y-1=0\) we can get \(x=1-y\)

Step 2: Substitute in the First Equation

Substitute \(x\) from equation 2 into equation 1, you get \(5(1-y)^2-2(1-y)y+5y^2-12=0\). Simplifying this equation yields \(6y^2-4y-7=0\)

Step 3: Solve the Quadratic Equation

To find the solution for \(y\) we solve the equation \(6y^2-4y-7=0\). Thus, the solutions are \(y= -\frac{1}{2}\) or \(y= \frac{7}{3}\)

Step 4: Substitute the Values of y in the Second Equation

Substitution of \(y\) values in equation \(x+y-1=0\), will give the corresponding \(x\) values. Thus for \(y= -\frac{1}{2}\) we have \(x=\frac{3}{2}\) and for \(y=\frac{7}{3} \) we have \(x=\frac{-4}{3}\)

Step 5: Verification

Plotting these equations on a graphical utility, it can be confirmed that indeed, the points of intersection are \((\frac{3}{2}, -\frac{1}{2})\) and \((\frac{-4}{3},\frac{7}{3})\)

Key Concepts

Quadratic Equations

Quadratic equations are polynomial equations of degree 2. They take the general form of . They are often presented in exercises where the goal is to find the values of the variable that make the equation true. Solving these equations usually involves factoring, completing the square, or applying the quadratic formula . In the provided example, the quadratic equation . Emerging after substitution is ( 6y^2-4y-7=0 the new quadratic equation to solve for the variable 'y'.

Substitution Method

The substitution method is a fundamental algebraic technique used to solve systems of equations where one equation is solved for one variable, and this expression is then substituted into the other equation. This process transforms the system into a single-variable equation that can be solved using standard algebraic methods.

In the exercise provided, the substitution method is applied by first isolating 'x' in the second equation, leading to the expression ( x=1-y ) and substituting it into the first equation. This direct substitution helps to eliminate one variable and simplifies the problem to solving for the remaining variable, 'y'. Once 'y' is determined, we can reverse the substitution to find the corresponding 'x' values.

Graphical Verification

Graphical verification is the process of confirming the solutions obtained algebraically by visually inspecting the graphs of the equations involved. This is done by plotting each equation on a coordinate plane and observing the points where the graphs intersect, which correspond to the solutions of the system.

For this exercise, a graphing utility can be used to plot the two equations, providing a visual affirmation of the intersection points calculated algebraically. The equation ( 5x^2-2xy+5y^2-12=0 ) and the linear equation could reveal the exact points of intersection as seen algebraically. Graphical verification serves not only as a confirmation but also aids in understanding the relationship between the equations and their solutions graphically.