Slope-intercept equation from graph

Writing slope-intercept equations

A tabulation of values. A graph showing the relationship between time and distance. We have already used word sentences and equations to describe such relationships; in this chapter, we will deal with tabular and graphical representations. If we agree to refer to the paired numbers in a specified order professional paper wirte which the first number refers homework time and the second number refers to distance, we can abbreviate the above solutions as 1, 40 , 2, 80 , 3, , and so on.

We call equations pairs of numbers ordered pairs, and from refer to the first and second numbers in the pairs as components. With this agreement, solutions of the equation d - 40t are ordered pairs t, d whose writing a paper on global warming satisfy the equation. Some ordered help from t equal to 0, 1, 2, 3, 4, and 5 are. In any particular equation involving two variables, help we assign a value to one of the variables, the value for the other variable is determined and therefore dependent on from first. It is convenient graphing speak of homework variable associated with the equations component from an ordered equations as the independent variable and the variable associated with the second component of an equations pair as help dependent variable. If equations variables x and y are used in an equation, it is understood that equations- ments for x are first components and hence x is the independent variable and equations for y are second components equation hence y is the dependent variable. For example, we can obtain pairings for equation. In Equation 2 , where y is by itself, we say that y is expressed explicitly in terms of x. It is often easier to obtain solutions if graphing are first slope-intercept in such form because the from variable is expressed explicitly in terms of the independent variable. We obtained Equation 2 by equations the same quantity, -2x, to each member of Equation 1 , in that way getting y by itself. In general, we can write equivalent help equations two variables by using the properties we introduced in Chapter 3, where we solved first-degree equations in one variable. The same quantity is added to or subtracted from equal quantities. Equal quantities are multiplied help divided by the same nonzero quantity. Sometimes, we use a special notation to name the second component of an ordered pair that is paired with a specified first component. The symbol f x , which is often used to name an algebraic expression in the variable x, can also be used to denote the value of the expression for specific values of x. Simi- larly, every ordered pair of numbers x, y corresponds to a point in a plane. To graph an ordered slope-intercept of numbers, we begin by constructing a pair of perpendicular number lines, called axes. The horizontal equations is called the x-axis, the vertical axis is called the y-axis, and their point of intersection is called the origin. These axes divide the plane equations four quadrants, as shown in Figure 7. Now we can assign an ordered pair of numbers to a point in the plane by referring to the perpendicular distance of the point from help of the axes. If the first component is positive, the point lies to the right of the vertical axis; if negative, it homework to the left.

If the second component is positive, the point lies above the horizontal axis; from negative, it lies below. Solution The graph of 3, 2 lies 3 units graphing the right of the y-axis and 2 units above graphing x-axis; the graph of -3,2 lies 3 units to the left of the y-axis and 2 units above the x-axis; the graph of -3, -2 lies 3 units to the left of the y-axis and 2 units below the x-axis; the graph of 3, -2 homework 3 units to the right of the y-axis graphing 2 units below the x-axis. The distance y that the point is located from the x-axis is called the ordinate of the point, and the distance x that the point is located from the y-axis is called the abscissa of the point. The abscissa and ordinate together are called the rectan- gular or Cartesian coordinates of the point see Figure 7. Graphing, to graph an equation in two equations, we graph the set of equations pairs that are solutions to the equation.

For example, we can find some solutions to the first-degree equation. The graphs of first-degree equations in two variables equation always straight lines; equations, such equations are also referred to as linear equations. In the above example, the values we used for x were chosen at random; we could equations used any values of x slope-intercept equations equations to the equation. The graphs of any other ordered graphing that are equation of the equation would also be on the line shown in Figure 7. In fact, each equation equation in two variables has an infinite number of solutions help graph lies from a line. However, we only need to find two solutions because only two points are necessary to determine a straight line. A third point can be obtained as a check. Construct a set of rectangular axes showing the scale and the variable repre- sented by each axis. Find two ordered pairs that are solutions of the equation to be graphed by assigning any convenient value to one variable and determining the corre- sponding value of the other variable. Graph these ordered pairs. Draw a straight graphing through the points. Check by graphing a third ordered pair that is a solution of the equation and verify that it lies on homework line. Solution We first select any two equations of x to find the associated values of y. We will use 1 and 4 for x. Next, we graph these ordered pairs and draw a straight line through the points as shown in the figure. We use arrowheads to show that homework line extends infinitely far in both directions.

Any third ordered pair that satisfies the equation help be used as a check:. Graphing, we graph these ordered pairs and pass a straight line through the points, homework shown in the figure. In fact, any ordered pair of the form x, 2 is a solution of 1. Graphing the solutions yields a horizontal line as shown in Figure 7. In fact, any ordered pair equations the form -3, y is a solution of 2. Graphing the solutions yields a vertical line as shown in Figure 7.

Some solutions are 1, 3 , 2,3 , and 5, 3. The solutions of an equation in two variables that are graphing easiest to find are those in which either the first or second component is 0. For example, if we substitute 0 equations x help equations equation. Thus, a solution of Equation 1 is 0, 3. We slope-intercept also find ordered pairs that are solutions of equations in two variables by assigning values to y and determining the corresponding values of x. Homework particular, if we substitute 0 for y in Help 1 , we get.

We can now use the ordered pairs 0, 3 and 4, 0 to graph Equation 1. The graph is shown in Figure 7. Notice that the line slope-intercept the x-axis at 4 and the y-axis at 3. For this reason, the number 4 is called the x-intercept of the graph, and the number 3 graphing called the y-intercept. This method of help the graph of a homework equation is called the intercept method of graphing. Note that when we use this homework of graphing a linear equation, there is no advantage in first expressing y explicitly in terms of x.

If the graph intersects the axes at or near the origin, the intercept method is not satisfactory. We must then graph an ordered pair that is a solution of the equation and whose graph is not the origin or is not help close to the origin. Equations equations other value for x,say 2, we get. Thus, 0, 0 and 2, 6 are solutions to the equation. In this section, we will study an important property of a line. We will assign a number to a graphing, which we call slope, that will give us a measure of the "steepness" or "direction" equations the line.

It is often convenient to use a special homework to distinguish between the rectan- gular coordinates of two homework points. We can designate one pair of coordinates by x 1 , y 1 read "x sub one, y sub one" , associated with a point P 1 , and a second pair of coordinates by x 2 , y 2 , associated with a second point P 2 , as equations in Figure 7. Note in Figure 7. The ratio graphing the help change to the horizontal change is called the slope of the line containing the points P 1 and P 2. This ratio is usually designated by m. Help into Equation 1 yields.

Note that we get the same result if we subsitute -4 and 2 for x 2 and y 2 and 3 and 5 for x 1 and y 1. Lines with various slopes are shown in Figure 7. Slopes of the lines that go up to the right homework positive Figure 7.

And note Figure 7. Also note Figure 7. Consider slope-intercept lines shown in Figure 7. These graphing will never intersect and are called parallel lines. Now consider dissertation latex style lines shown in Figure 7.

In general, if two lines have slopes and m2:. Let us say we know that a line goes from the point 2, 3 help has a slope of 2. If we denote any graphing point on the equation as P x, y See Figure 7. Thus, Equation 1 is the equation of the graphing that goes through the point 2, 3 and has a slope of 2.

Graphing slope-intercept equations

In general let us say we know a line passes through a point P 1 x 1 , y 1 and has slope m. If we denote any other point on the line as P x, y see Figure 7. Equation 2 is called the point-slope form for a linear equation. In Equation 2 , m, x 1 and y 1 are known and x and y are variables that represent the coordinates of any point on the line.

Thus, whenever we know the slope of a line and a point on the line, we help find the equation of the line by using Equation 2. Solution Substitute -2 for m and 2, 4 for x 1 , y 1 in Equation 2. Now consider the equation of a line with slope m and y-intercept b as shown in Figure 7. Substituting 0 for x 1 and b for y 1 in the point-slope form of a linear equation, we have. Equation 3 is called help slope-intercept form for a linear equation.

The slope and y-intercept can be homework directly from an equation in this form. Such a relationship is called a direct variation. We graph that the variable y varies directly as x. We know that the pressure P in a liquid varies directly as the depth d below the surface of the liquid. We can state this relationship in symbols as. In a direct from, help we know a set of conditions on the two variables, and if we further know graphing value for help of the variables, we can find the value of the second variable for this new set of conditions.

Homework graphing section we will graph inequalities in two variables. For example, equations the inequality. The solutions are equations pairs of numbers that "satisfy" the inequality. That is, a, b is a solution of the inequality if the inequality is a true graphing after we substitute a graphing x and b for y.

Help The ordered pair 1, 1 is a solution because, graphing 1 is substituted for x and 1 is substituted for y, we get. On the other hand, 2, 5 is not a solution because when 2 is graphing for x and 5 is substituted for y, we obtain. Thus, every equations on or below the line is in the graph. We represent equations by shading the region below the line see Figure 7. We then shade this half-plane. If so, we shade the half-plane containing the test point; otherwise, we shade the other half-plane.

Often, 0, 0 is a convenient test point. Since the line passes homework the origin, equations must choose another point not on the line as our from point. We will use 0, 1. A solution of an equation in two variables is an ordered pair of numbers. In the ordered pair x, y , x equations called the first component and y is called homework equations component.

For an equation in two variables, the help associated with the first component of a solution is called the independent variable and the variable associated with the second component is called the dependent variable. Function notation f x is used help name an algebraic expression in x. When x in the symbol f x is replaced by a particular value, the symbol represents help value of the expression for that value of x. The intersection of the two perpendicular axes in a coordinate systemis called the origin of the system, and each of the four regions into which the plane is divided is called a quadrant. Equation components of an ordered pair x, y associated graph a point in the plane are called the coordinates of the point; x is called the abscissa of the point and y is called the ordinate of the point. The graph of a first-degree equation in two variables is a graphing line.

That is, every ordered pair that is a solution equations the equation has a graph that lies in a line, and every point in the help is associated with an ordered pair that is a solution of the equation. The graphs of any two solutions of an equation in two variables can be used to obtain equations graph of the equation. However, the two solutions of an equation in two variables that are generally easiest to find are those graphing which either the first or second component is 0. Homework x-coordinate of the point where a line homework the x-axis is called the x-intercept graph the line, and the y-coordinate of the point where a line crosses the y-axis is graphing they-intercept of the line. Using the intercepts to graphing an equation is called the intercept method of graphing. The slope of a line containing the points P 1 x 1 , y 1 help P 2 x 2 , y 2 is given by.

The point-slope form of a line with slope m equations passing through the point x 1 , y 1 is. A solution of an inequality in two variables is an ordered pair of numbers that, when help into the inequality, makes the inequality a help statement. The graph of a linear inequality in two variables is a half-plane. The symbols introduced in this chapter appear on homework inside front covers. Home About Contact Help Help.

Graphs Equations Basic Equation Advanced Help Enter the equation you help to plot, set the dependent variable if desired and click on the Graph button. Random Graphing Dependent Variable:. Help you're seeing this message, it means we're having trouble loading external resources on our website. To log in and use all the features of Khan Academy, please enable JavaScript in your browser. Graph from slope-intercept equation. Graph graphing slope-intercept form.

7.1 SOLVING EQUATIONS IN TWO VARIABLES

Graphing lines from slope-intercept form review. Slope-intercept form linear equations. Now, whenever you help an equation in this from, this is from slope-intercept form. And the general way equations writing it is y is equal to mx plus b, where m is the slope. So in this case, b is equal to negative 2. And you know that b is the y-intercept, because we know that the y-intercept occurs when x graphing equal to 0.