Problem 64 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solve the system graphically or algebraically. Explain your choice of method. $$\left\\{\begin{array}{l} x^{2}+y^{2}=25 \\ 2 x+y=10 \end{array}\right.$$

Step-by-Step Solution

Answer

The solutions to the system of equations are (4,2) and (3,4)

Step 1: Rearrange Line Equation

To use the substitution method in solving the system of equations, the second equation can be rearranged to express \(y\) in terms of \(x\). This is done by subtracting \(2x\) from both sides of the equation to get \(y = 10 - 2x\)

Step 2: Substitute Line Equation into Circle Equation

Replace \(y\) in the circle equation with \(10 - 2x\) to get \(x^2 + (10 - 2x)^2 = 25\)

Step 3: Simplify and Solve for X

Expand and simplify the equation to get \(x^2 + 100 - 40x + 4x^2 = 25\), then combine like terms to get \(5x^2 - 40x + 75 = 0\). Use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) to solve for \(x\), giving \(x = 4\) or \(x = 3\)

Step 4: Solve for Y

Substitute the solutions for \(x\) found earlier into the line equation \(y = 10 - 2x\) to find the corresponding \(y\)-values. This gives \(y = 2\) when \(x = 4\) and \(y = 4\) when \(x = 3\)