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Chapter 8: Problem 86

Solve the system \(y=x^{2}+1\) and \(y=x+1\).

Short Answer

Expert verified
The solutions are (0, 1) and (1, 2).

Step by step solution

01

Set the Equations Equal

Since both equations are equal to y, set the two equations equal to each other: \[x^2 + 1 = x + 1\]
02

Simplify the Equation

Subtract x and 1 from both sides of the equation to bring all the terms to one side: \[x^2 + 1 - x - 1 = 0\]This simplifies to: \[x^2 - x = 0\]
03

Factor the Equation

Factor out the common term x from the simplified equation: \[x(x - 1) = 0\]
04

Solve for x

Set each factor equal to zero to find the values of x: \[x = 0\]\[x - 1 = 0\] \[x = 1\]Thus, the solutions for x are 0 and 1.
05

Find Corresponding y Values

Use the original equation \(y = x + 1\) to find the corresponding y values for each x: For \(x = 0\): \[y = 0 + 1 = 1\]For \(x = 1\): \[y = 1 + 1 = 2\]
06

Write the Solution as Points

The solutions for the system as points are: \((0, 1)\) and \((1, 2)\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Quadratic Equations
A quadratic equation is a type of polynomial equation of degree 2. It has the general form: \(ax^2 + bx + c = 0\), where a, b, and c are constants and \(a eq 0\). The solution to a quadratic equation can be found using various methods including: factoring, completing the square, or using the quadratic formula. The graph of a quadratic equation is a parabola which can open upwards or downwards depending on the sign of the leading coefficient (a). In our problem, the equation \(y = x^2 + 1\) is quadratic and is one of the main equations we need to solve.
Linear Equations
A linear equation is a polynomial equation of degree 1. It has the general form \(y = mx + b\), where m is the slope, and b is the y-intercept. This type of equation graphs as a straight line. In our exercise, the linear equation provided is \(y = x + 1\). Solving systems that involve both quadratic and linear equations typically requires finding the points where their graphs intersect. These points of intersection provide the solutions to the system.
Factoring
Factoring is a method used to break down polynomials into simpler 'factor' polynomials whose product is the original polynomial. For instance, given the polynomial \(x^2 - x = 0\), we can factor out the common term x to get \(x(x - 1) = 0\). Factoring is beneficial because it simplifies the process of solving equations. After factoring, we set each factor equal to zero to find the solutions. In our case, we find that \(x = 0\) and \(x = 1\) are the solutions.

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Most popular questions from this chapter

Write a dependent system of two linear equations for which \(\\{(t, t+5) | t \text { is any real number }\\}\) is the solution set. Ask a classmate to solve your system.

Sketch the graph of the solution set to each nonlinear system of inequalities. $$\begin{aligned}&y>|2 x|-4\\\&y \leq \sqrt{4-x^{2}}\end{aligned}$$

Solve each problem using two variables and a system of two equations. Solve the system by the method of your choice. Note that some of these problems lead to dependent or inconsistent systems. The rancher's wife raises snakes and white mice. One day she observed that her animals have a total of 78 eyes and 38 tails. How many animals of each type does she have?

Solve each system. $$\begin{aligned} x+y+2 z &=7.5 \\ 3 x+4 y+z &=12 \\ 5 x+2 y+5 z &=21 \end{aligned}$$

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