## Elementary Linear Algebra 7th Edition

Published by Cengage Learning

# Chapter 1 - Systems of Linear Equations - 1.1 Introduction to Systems of Linear Equations - 1.1 Exercises - Page 10: 1

#### Answer

Yes, the equation is linear with 2 variables, $x$ and $y$

#### Work Step by Step

Method #1: A linear equation has the following characteristics: $$a_{1}x_{1}+a_{2}x_{2}+...a_{n}x_{n}=b$$ Where $a$ and $b$ are real numbers and $x$ represents the variables. Since our equation follows this format with 2 variables it can be deduced to be a linear equation with 2 variables. Method #2: By using graphical methods, one can observe no curvature in the line, thus expressing it as a linear equation. Otherwise one can make a table, observing a linear and constant change. Method #3: If you convert to standard form by solving for $y$: $$2x-3y=4$$ 1: Subtract $2x$ from both sides: $-3y=4-2x$ 2: Reorder $4-2x$ into $-2x+4$: $-3y=-2x+4$ 3: Invert positive and negative: $3y=2x-4$ 4: Divide both sides by $3$ for: $y=\frac{2}{3}x-\frac{4}{3}$ Now you can see the equation is in standard form: $y=mx+b$ With $\frac{2}{3}$ as the slope ($m$) and $-\frac{4}{3}$ as the $y$ intercept ($b$). This form is strictly linear, there for identifies this equation as linear with 2 variables, $x$ and $y$. Method #4: You can verify it is linear with two variables by checking the x and y intercepts. All linear lines will have two variables (often x and y) and will have only one x-intercept and one y-intercept. An intercept is when one variable is set to zero and the other shows a value. That value can be in the x or in the y place in the coordinate writing of points. For example, when x is set to zero, we call the resulting y value a y-intercept and plot it on the y-axis. Likewise, when y is set to zero, we call the resulting x value a x-intercept and plot it on the x-axis. You can solve for these in equations, by setting the x variable to zero and then completing the rest of the functions or by setting the y variable to zero and then solving for x. Example #1: Solve for the x-intercept with the following equation: y=$2x + 2$ $0 = 2x + 2$ Subtract 2 from both sides to begin undoing the equation and solving for x. $-2 = 2x$ Then, divide by 2 on both sides to isolate the variable x and find its value. $-1= x$ -1 is the resulting x-intercept and can be shown as a point on a coordinate plane like this: (-1,0). Example #2: Solve for the y-intercept with the following equation: y= $2x + 2$ Now, we begin by setting x to zero. $y= 2*0 +2$ First, we use PEMDAS or the order of operations, and determine that we should multiply 2 and 0 together before adding 0 and 2. $y = 0 + 2$ Next, we complete the addition to get the value for the y-intercept. $y= 2$ 2 is the value for the y-intercept, and that can be shown as a point on a coordinate plane in this form: (0,2). As the line we tested only has two intercepts, it is linear with two variables. You can apply these same principles, as well as the principles above, to any line you are testing to see if it is linear with two variables.
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