Which of the following are linear equations in one variable? a) y^2+8 y+15=0 b) (1)/(2) w-5(3 w+

Which of the following are linear equations in one variable?...

Key Concepts

Constants and Coefficients

In the context of linear equations, constants and coefficients play key roles. The coefficient of the variable indicates the slope of the line when the equation is graphed, while the constant term represents the y-intercept. Recognizing these components helps in understanding the overall behavior and properties of the equation.

Linear Equations in One Variable

A linear equation in one variable is an algebraic equation where the variable appears only to the first power and is not multiplied by itself. The standard form of such an equation is ax + b = c, where a, b, and c are constants and a is nonzero. This type of equation represents a straight line when graphed on a coordinate plane.

Degree of an Equation

The degree of an equation is determined by the highest exponent of the variable present. For an equation to be classified as linear, the variable must only be raised to the power of one. If the variable appears with an exponent higher than one (such as a squared term), the equation becomes non-linear.

Algebraic Simplification

Simplification is the process of combining like terms and using the distributive property to rewrite equations in a simpler form. This process is essential for identifying the true structure of an equation, particularly to verify whether it is linear by ensuring that the variable only has an exponent of one.

Transcript

00:01 Hi everyone.

00:02 So we have these four expressions slash equations in front of us, and we want to know which of them are linear equations in one variable.

00:10 Now, what is a linear equation in one variable? it's any equation that could be written out like this.

00:17 Ax plus b equals 0.

00:22 And then, hold on, sorry.

00:26 And that's given that a does not equal 0.

00:32 So what does this mean? so we have one variable, hence the single variable linear equation, and then we have a coefficient in front of it, and then we have one single constant, and then all of this is equal to zero.

00:54 Or another way to write it would be a x equals negative b.

00:58 That b looks terrible, sorry.

01:02 So let's go down the list here, and see which of these meet our conditions.

01:08 So the first one, y squared plus 8y plus 15 equals 0.

01:12 This is not a single variable or linear equation in one variable because we have this y squared term right here.

01:22 So that prevents us from being able to just write it out in a x plus b.

01:26 What we have here is a x squared plus bx plus c, which is not the same thing as a x plus b.

01:37 So, so a is a no.

01:41 So for part b, sorry, one second, let's make that better.

01:45 There we go.

01:46 We have one half w minus five times three w plus one equals six.

01:50 Now this is a single variable linear equation because we can simplify our terms here.