Solve the System of Equations y = square root of x-7 y=-x-1

Solution:

Step:1

Combine the equations by setting them equal to each other since they both equal $y$.

$\sqrt{x} - 7 = -x - 1$

Step:2

Isolate $x$ in the equation $\sqrt{x} - 7 = -x - 1$.

Step:2.1

Shift all terms without $\sqrt{x}$ to the opposite side.

Step:2.1.1

Add $7$ to both sides.

$\sqrt{x} = -x + 6$

Step:2.2

Square both sides to eliminate the square root.

$(\sqrt{x})^2 = (-x + 6)^2$

Step:2.3

Simplify both sides of the equation.

Step:2.3.1

Express $\sqrt{x}$ as $x^{\frac{1}{2}}$.

$(x^{\frac{1}{2}})^2 = (-x + 6)^2$

Step:2.3.2

Simplify the left side by applying the power rule $(a^m)^n = a^{mn}$.

$x = (-x + 6)^2$

Step:2.3.3

Expand the right side.

$x = (-x + 6)(-x + 6)$

Step:2.3.3.1

Distribute and combine like terms.

$x = x^2 - 12x + 36$

Step:2.4

Solve the resulting quadratic equation.

Step:2.4.1

Move all terms to one side.

$x^2 - 13x + 36 = 0$

Step:2.4.2

Factor the quadratic.

$(x - 9)(x - 4) = 0$

Step:2.4.3

Find the roots by setting each factor equal to zero.

$x = 9$ or $x = 4$

Step:2.4.4

Discard any extraneous solutions that do not satisfy the original equation.

$x = 4$

Step:3

Determine the value of $y$ when $x = 4$.

$y = -x - 1$ $y = -4 - 1$ $y = -5$

Step:4

Combine the values of $x$ and $y$ to form the solution to the system.

The solution is $(4, -5)$.

Step:5

Present the solution in different formats.

Point Form: $(4, -5)$ Equation Form: $x = 4, y = -5$

Knowledge Notes:

1. Square Root Function: The square root function, denoted as $\sqrt{x}$, is the inverse operation of squaring a number. To solve equations involving square roots, one often squares both sides to eliminate the radical.

2. Solving Quadratic Equations: A quadratic equation is typically in the form $ax^2 + bx + c = 0$. It can be solved by factoring, completing the square, or using the quadratic formula. Factoring involves finding two numbers that multiply to give $ac$ and add to give $b$.

3. Extraneous Solutions: When both sides of an equation are squared, extraneous solutions may be introduced. These are solutions that do not satisfy the original equation. It is important to check all potential solutions in the original equation.

4. Systems of Equations: A system of equations is a set of two or more equations with the same variables. The solution to a system of equations is the set of values that satisfy all equations simultaneously. Systems can be solved by various methods, including substitution, elimination, and graphing.

5. Radical Equations: Equations that contain a variable within a radical. To solve these equations, one typically isolates the radical on one side and then raises both sides of the equation to the power of the radical to eliminate it. It is important to check for extraneous solutions when solving radical equations.