           Linear Equations

A linear equation looks like any other equation. It is made up of two expressions set equal to each other. A linear equation is special because:

1. It has one or two variables.
2. No variable in a linear equation is raised to a power greater than 1 or used as the denominator of a fraction.
3. When you find pairs of values that make the linear equation true and plot those pairs on a coordinate grid, all of the points for any one equation lie on the same line. Linear equations graph as straight lines.

A linear equation in two variables describes a relationship in which the value of one of the variables depends on the value of the other variable. In a linear equation in x and y, x is called x is the independent variable and y depends on it. We call y the dependent variable. If the variables have other names, yet do have a dependent relationship, the independent variable is plotted along the horizontal axis. Most linear equations are functions (that is, for every value of x, there is only one corresponding value of y). When you assign a value to the independent variable, x, you can compute the value of the dependent variable, y. You can then plot the points named by each (x,y) pair on a coordinate grid. The real importance of emphasizing graphing linear equations with your students, is that they should already know that any two points determine a line, so finding many pairs of values that satisfy a linear equation is easy: Find two pairs of values and draw a line through the points they describe. All other points on the line will provide values for x and y that satisfy the equation.

Describing Linear Relationships
The graphs of linear equations are always lines. One important thing to remember about those lines is: Not every point on the line that the equation describes will necessarily be a solution to the problem that the equation describes.

Examples of Linear Relationships

• distance = rate time

In this equation, for any given steady rate, the relationship between distance and time will be linear. However, distance is usually expressed as a positive number, so most graphs of this relationship will only show points in the first quadrant. Notice that the direction of the line in the graph below is from bottom left to top right. Lines that tend in this direction have positive slope. A positive slope indicates that the values on both axes are increasing from left to right. • amount of water in a leaky bucket = rate of leak time

In this equation, since you won't ever have a negative amount of water in the bucket, the graph will also show points only in the first quadrant. Notice that the direction of the line in this graph is top left to bottom right. Lines that tend in this direction have negative slope. A negative slope indicates that the values on the y axis are decreasing as the values on the x axis are increasing. • number of angles of a polygon = number of sides of that polygon

Again in this graph, we are relating values that only make sense if they are positive, so we show points only in the first quadrant. In this case, since no polygon has fewer than 3 sides or angles, and since the number of sides or angles of a polygon must be a whole number, we show the graph starting at (3,3) and indicate with a dashed line that points between those plotted are not relevant to the problem. • degrees Celsius = 5/9 (degrees Fahrenheit – 32)

Since it's perfectly reasonable to have both positive and negative temperatures, we plot the points on this graph on the full coordinate grid.

•        Slope        Two-step Linear Equations
with Rational Numbers. Slope
• Steepness and Direction

The slope of a line tells two things: how steep the line is with respect to the y axis and whether the line slopes up or down when you look at it from left to right. When you're plotting data, slope tells you the rate at which the dependent variable is changing with respect to the change in the independent variable. This gives you a valuable clue about how to find slope: Pick any two points on the line. To find how fast y is changing, subtract the y value of the second point from the y value of the first point (y2y1). To find how fast x is changing, subtract the x value of the second point from the x value of the first point (x2x1). To find the rate at which y is changing with respect to the change in x, write your results as a ratio: (y2y1)/(x2x1). If we designate Point A as the first point and Point B as the second point, the slope of the line is ( 2 – 4)/( 1 – 2) = 6/ 3. or 2. It does not matter which point you designate as point 1, just as long as you use the same point as the first point when calculating change in y and change in x. If we designate Point B as the first point and Point A as the second point, the value of the slope is the same: (4 – 2)/(2 – 1) = 6/3, or 2. It is also the same value you will get if you choose any other pair of points on the line to compute slope.

• Slope-Intercept Form

The equation of a line can be written in a form that gives away the slope and allows you to draw the line without any computation. If students are comfortable with solving a simple two-step linear equation, they can write linear equations in slope-intercept form. The slope-intercept form of a linear equation is y = mx + b. In the equation, x and y are the variables. The numbers m and b give the slope of the line (m) and the value of y when x is 0 (b). The value of y when x is 0 is called the y-intercept because (0,y) is the point at which the line crosses the y axis. You can draw the line for an equation in this form by plotting (0,b), then using m to find another point. For example, if m is 1/2, count 2 on the x axis, then 1 on the y axis to get to another point (1, b + 2) The equation for this line is y + 3 = 2x. In slope-intercept form, the equation is y = 2x – 3. You can see that the slope m = 2 and the slope really is 2 since for every 2 change in y, there is a 1 change in x. Now look at b in the equation: 3 should be the y value where x = 0 and it is.

• Positive Slope

When a line slopes up from left to right, it has a positive slope. This means that a positive change in y is associated with a positive change in x. The steeper the slope, the greater the rate of change in y in relation to the change in x. When you are dealing with data points plotted on a coordinate plane, a positive slope indicates a positive correlation and the steeper the slope, the stronger the positive correlation.

Consider gas mileage. If you drive a big, heavy, old car, you get poor gas mileage. The rate of change in miles traveled is low in relation to the change in gas consumed, so the value m is a low number and the slope of the line is fairly gradual. If you drive a light, efficient car, you get better gas mileage. The rate of change in the number of miles you travel is higher in relation to the change in gas consumed, so the value of m is a greater number and the line is steeper. Both rates are positive, because you still travel a positive number of miles for every gallon of gas you consume.

• • Negative Slope

When a line slopes down from left to right, it has a negative slope. This means that a negative change in y is associated with a positive change in x. When you are dealing with data points plotted on a coordinate plane, a negative slope indicates a negative correlation and the steeper the slope, the stronger the negative correlation.

Consider working in your vegetable garden. If you have a flat of 18 pepper plants and you can plant 1 pepper plant per minute, the rate at which the flat empties out is fairly high, so the absolute value of m is a greater number and the line is steeper. If you can only plant 1 pepper plant every 2 minutes, you still empty out the flat, but the rate at which you do so is lower, the absolute value of m is low, and the line is not as steep.

• • Zero Slope

When there is no change in y as x changes, the graph of the line is horizontal. A horizontal line has a slope of zero. • Undefined Slope

When there is no change in x as y changes, the graph of the line is vertical. You could not compute the slope of this line, because you would need to divide by 0. These lines have undefined slope. • Lines with the Same Slope

Lines with the same slope are either the same line, or parallel lines. In all three of these lines, every 1-unit change in y is associated with a 1-unit change in x. The rate of change is 1/1. All three have a slope of 1.

Solving Two-Step Linear Equations with Rational Numbers

When a linear equation has two variables, as it usually does, it has an infinite number of solutions. Each solution is a pair of numbers (x,y) that make the equation true. Solving a linear equation usually means finding the value of y for a given value of x.

• When the Equation is Already in Slope-Intercept Form

If the equation is already in the form y = mx + b, with x and y variables and m and b rational numbers, then the equation can be solved in algebraic terms. To find ordered pairs of solutions for such an equation, choose a value for x, and compute to find the corresponding value for y. You'll notice that the easiest value to choose for x is often 0, because in that case, y = b. Students may be asked to make tables of values for linear equations. These are simply T-tables with lists of values for x with the corresponding computed values for y.

Two-step equations involve finding values for expressions that have more than one term. The terms in an expression are separated by addition or subtraction symbols. 2x is an expression with one term. 2x + 6 has two terms. To find a value for y given a value for x, substitute the value for x into the expression and compute. First, find the value of the term that contains x, then find the value of the entire expression.

Consider the equation y = 2x+ 6. Find the value for y when x = 5.
 First, substitute the value for x into the equation .; y = 2(5) + 6 Then, find the value of the term that contains x. y = 10 + 6 Last, find the value of the entire expression. y = 16

• When the Equation is Not in Slope-Intercept Form

When a linear equation is not in slope-intercept form (y = mx + b), students can still make a table of values to find solutions for the equation, but it may be simpler to put the equation in slope-intercept form first. This requires mirroring operations (balancing) on each side of the equation until y is by itself on the one side of the equation, set equal to an expression involving x. You can manipulate the equation in this way because of the equality properties:

If a = b, then a + c = b + c
If a = b, then ac = bc
If a = b, then ac = bc
If a = b, then a ÷ c = b ÷ c

Consider 2x + y – 6 = 0. This equation is not in slope-intercept form. There are two ways to put it in slope-intercept form.

 1. Show the original equation. 2x + y – 6 = 0 Subtract y from each side. 2x + y – y – 6 = 0 – y 2x – 6 = 0 – y Multiply each side by 1. 1(2x – 6) = 1(y) 2x + 6 = y

 2. Show the original equation. 2x + y – 6 = 0 Add 6 to each side. 2x + y – 6 + 6 = 0 + 6 2x + y = 6 Subtract 2x from each side. 2x – 2x + y = 6 – 2x y = 6 – 2x

The two equations, 2x + 6 = y and y = 6 – 2x are equivalent because you can turn one into the other by using the symmetric property of equality, which states that if a = b, then b = a and the commutative property, which states that a + b = b + a.
 commutative property 2x + 6 = y 6 – 2x = y symmetric property 6 – 2x = y y = 6 – 2x

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