Question

State whether or not the given equation is linear. $$x_{1}+y+t=1$$

Short answer

The equation is linear.

Step-by-step solution

Identify Variables and Constants

Examine the given equation \(x_1 + y + t = 1\). Here, \(x_1, y,\) and \(t\) are variables, and \(1\) is a constant.

Check the Power of Each Variable

In a linear equation, each variable should have a power of 1. In the equation \(x_1 + y + t = 1\), the powers of \(x_1, y,\) and \(t\) are all 1.

Check for Products of Variables

Ensure there are no products of variables within the equation. The equation \(x_1 + y + t = 1\) contains no terms that are the product of two or more variables.

Confirm the Linear Form

For a linear equation, the general form is \(a_1x_1 + a_2x_2 + \, ... \, + a_nx_n = b\), where each \(x_i\) is a variable, \(a_i\) are constants, and \(b\) is a constant. The given equation matches this form.

Conclusion: Determine Linearity

The equation \(x_1 + y + t = 1\) is a linear equation because each variable is to the first power, there are no products or functions of variables, and it is in a standard linear form.

Key concepts

Variables and Constants

In mathematics, variables are symbols that represent unknown values or quantities that can change. They are usually denoted by letters such as \(x\), \(y\), and \(t\). In the equation \(x_1 + y + t = 1\), \(x_1\), \(y\), and \(t\) are the variables. They stand for different quantities we might want to calculate or solve for.
In contrast, constants are fixed values that do not change. They provide specific, consistent numerical information within equations. In our equation, the number \(1\) is a constant. It tells us exactly what the sum of \(x_1\), \(y\), and \(t\) should equal.

• Variables can take different values.
• Constants remain the same across calculations or problems.

Understanding these concepts is crucial for recognizing and working with any type of equation.

Power of Variables

In order for an equation to be classified as linear, each variable must have a power, or exponent, of 1. This property ensures the graph of the equation is a straight line.
In the exercise equation, \(x_1 + y + t = 1\), each of the variables \(x_1\), \(y\), and \(t\) has an implicit power of 1, even though the exponents are not written. Exponents are typically omitted when they equal 1, but it's important to remember they dictate the equation's linearity. Here’s why this matters:

• The power of 1 keeps the relationship between variables straightforward.
• Any higher powers (like squared terms) would make the equation non-linear.

Therefore, recognizing the power of variables helps us verify whether we are dealing with a linear relationship or not.

General Form of Linear Equations

Linear equations have a specific format, known as the general form. This is often written as \(a_1x_1 + a_2x_2 + \, ... \, + a_nx_n = b\), where:

• \(a_1, a_2, ..., a_n\) are coefficients, which are constants that multiply each variable.
• \(x_1, x_2, ..., x_n\) are variables.
• \(b\) is a constant.

Our example equation \(x_1 + y + t = 1\) fits this general form, making it a clear-cut linear equation:

• The coefficients of \(x_1\), \(y\), and \(t\) are all implicitly \(1\), assuming no coefficients suggest a multiplicative identity (i.e., multiplication by 1).
• \(b\) is \(1\).

Knowing the general form allows us to quickly determine whether an equation is linear, by confirming each part aligns with this layout. It provides a useful template to categorize equations efficiently.