Question

Solve the system by elimination. \(y=x^2+4 x+7\) \(-y=4 x+7\)

Short answer

The possible solutions for the given system are \((-4 + \sqrt{2}, -1 - 4\sqrt{2})\) and \((-4 - \sqrt{2}, -1 + 4\sqrt{2})\).

Step-by-step solution

Add the equations together to eliminate one variable

Add \(y = x^2+4x+7\) and \(-y = 4x+7\) to eliminate y. The result is \(0 = x^2+4x+7 + 4x+7\). This simplifies to \(0 = x^2+8x+14\).

Solve the equation

To solve the equation \(0 = x^2+8x+14\), first move the 14 to the other side of the equation, to get \(x^2+8x=-14\). Then complete the square: half the coefficient of x is 4, and its square is 16. Adding 16 to both sides gives \(x^2+8x+16 = 2\) Simplify further to get \((x+4)^2 = 2\), then take the square root of both sides to find that \(x+4=\pm \sqrt{2}\). Solve for x by subtracting 4 from both sides, which gives \(x = -4 \pm \sqrt{2}\).

Substitute x into one of the original equations to find y

Substitute \(x = -4 + \sqrt{2}\) into the equation \(y = x^2+4x+7\) to get \(y =(-4 + \sqrt{2})^2 +4(-4 + \sqrt{2}) +7\), simplifying to \(y =-1 - 4\sqrt{2}\). Similarly, substituting \(x = -4 - \sqrt{2}\) into the same equation gives \(y =-1 + 4\sqrt{2}\). There are thus two solutions: \((-4 + \sqrt{2}, -1 - 4\sqrt{2})\) and \((-4 - \sqrt{2}, -1 + 4\sqrt{2})\).

Key concepts

Elimination Method

The elimination method is a powerful technique used to solve systems of equations, particularly when you have two equations and two unknown variables. The goal in this method is to eliminate one of the variables by combining the equations. In the original exercise, we eliminated the variable \( y \) by adding the two equations together. This resulted in a simplified equation in terms of \( x \) only. By strategically adding or subtracting equations, you can cancel out one of the variables. This effectively reduces a system of equations to a single equation, which is often easier to solve.

Solving Quadratics

Quadratic equations are polynomial equations of the form \( ax^2 + bx + c = 0 \). Solving quadratics is a standard process in algebra that typically involves factoring, using the quadratic formula, or completing the square. In the exercise, we obtained a quadratic equation \( x^2 + 8x + 14 = 0 \) after using the elimination method. To solve for \( x \), transforming the quadratic equation into a simpler form is essential for finding solutions. This often lays the groundwork for further techniques like completing the square or using the quadratic formula.

Completing the Square

Completing the square is a method used to solve quadratic equations by transforming them into a perfect square trinomial. A perfect square trinomial is an expression of the form \((x + p)^2 = q\). In the solution, the quadratic \( x^2 + 8x + 14 = 0 \) was rearranged as \( x^2 + 8x = -14 \). By taking half of the coefficient of \( x \) (which is 4), squaring it to get 16, and adding this to both sides of the equation, we form the perfect square trinomial \((x + 4)^2 = 2\). Taking the square root of both sides and solving for \( x \) provides the solutions for the variable.

Algebraic Substitution

Algebraic substitution is a technique used to find the values of remaining variables once a certain solution is found. After solving for \( x \) using methods like completing the square, substitution is used to find \( y \). In the exercise, the values of \( x \) were \( -4 \pm \sqrt{2} \). These values were substituted back into one of the original equations, \( y = x^2 + 4x + 7 \), to find the corresponding \( y \) values. This step ensures all variables are accounted for and gives the complete solutions for the system of equations. The substitution process thus connects the solutions from one part of the system to the results in the other.