Question

\(y=5+x, y=x^{2}, x=0\), and \(x=1\)

Short answer

The values of y for x=0 are 5 (linear) and 0 (quadratic); for x=1, they are 6 (linear) and 1 (quadratic).

Step-by-step solution

Understand the Equations

We are given two equations to work with: 1. A linear equation: \(y = 5 + x\)2. A quadratic equation: \(y = x^2\). We want to find the value of \(y\) using both equations, first with \(x = 0\) and then with \(x = 1\).

Substitute x = 0 into Equations

Let's substitute \(x = 0\) into the linear equation:\[y = 5 + 0 = 5\]Next, substitute \(x = 0\) into the quadratic equation:\[y = 0^2 = 0\]So, when \(x = 0\), we get two different values for \(y\): 5 and 0.

Substitute x = 1 into Equations

Now, substitute \(x = 1\) into the linear equation:\[y = 5 + 1 = 6\]Then, substitute \(x = 1\) into the quadratic equation:\[y = 1^2 = 1\]So, when \(x = 1\), we get two different values for \(y\): 6 and 1.

Analyze the Results

For \(x = 0\), the linear and quadratic equations give \(y = 5\) and \(y = 0\) respectively.For \(x = 1\), the linear and quadratic equations give \(y = 6\) and \(y = 1\) respectively.This shows that for each value of \(x\), the two equations yield different values for \(y\).

Key concepts

Linear Equations

Linear equations are a fundamental part of algebra and mathematics. A linear equation is a statement of equality between two algebraic expressions that form a straight line when plotted on a graph.

They can be represented in a general form as \( y = mx + b \), where:

• \( m \) is the slope, indicating the steepness or incline of the line.
• \( b \) is the y-intercept, the point where the line crosses the y-axis.

Linear equations create a straight line, with no curves, when visualized on a graph. The equation \( y = 5 + x \) is an example, with a slope of 1 and y-intercept of 5. When swapping different values of \( x \) into the equation, the result gives a different value for \( y \) that places the point along the line. In the exercise, we placed \( x = 0 \) and \( x = 1 \) into our linear equation to study the corresponding changes in \( y \). Linear equations are versatile tools in analyzing relationships within algebra, making them essential for solving practical everyday problems.

Quadratic Equations

Quadratic equations differ significantly from linear ones, offering a parabolic curve on a graph. These equations are of the form \( y = ax^2 + bx + c \). Typically, their solutions involve curves, which can either open upward or downward depending on the sign of \( a \). In simpler terms:

• A positive \( a \) leads to a U-shaped curve.
• A negative \( a \) leads to an upside-down U-shape.

In our exercise, the equation \( y = x^2 \) represents a basic quadratic equation where \( a = 1 \), \( b = 0 \), and \( c = 0 \). For this equation, only the term \( x^2 \) influences the curve's shape. Substituting \( x \) with values like 0 and 1 changes \( y \, \)'s location on this curve. As shown: \( y = 0^2 = 0 \) and \( y = 1^2 = 1 \) illustrate the quadratic equation's outcome for these specific \( x \) values. Quadratic equations provide insights into more complex relationships, motion, and the trajectory of objects, playing a role in diverse areas of science and engineering.

Substitution Method

The substitution method is one of the main strategies for solving systems of equations involving variables. It revolves around replacing one variable in an equation with another expression. This technique is especially useful when dealing with systems that mix different equations like linear and quadratic types.

Here’s a simple guide to help understand how it's applied:

• First, solve one of the equations for one of the variables in terms of the others.
• Substitute this expression into the other equation(s), reducing the number of variables.
• Simplify and solve the resulting equation, substituting back if necessary for the other variable(s).

In the exercise, we applied the substitution method by inserting specific values of \( x \), like 0 and 1, into both equations to find corresponding \( y \) values. This helped illustrate the ​differences in outputs when same \( x \,\) values were used. This method helps solve systems of equations efficiently, especially when one equation simplifies easily, providing clarity in situations where equations may seem complex or abstract at first glance.