Graph X-2y 0

[Marthe Bernskoetter]

Tograph the plane x+2y+3z=0, you can use the intercept method. This involves finding the x, y, and z intercepts of the plane and connecting them to form a triangle. Alternatively, you can use a three-dimensional graphing calculator or software to plot the plane.

The equation x+2y+3z=0 represents a plane in three-dimensional space. It is a linear equation, meaning that it forms a straight line when graphed. The coefficients of the variables x, y, and z determine the slope of the plane.

The graph plane x+2y+3z=0 is related to the Cartesian coordinate system because it is a representation of a two-dimensional plane in a three-dimensional space. The Cartesian coordinate system is used to plot points and graph equations in three dimensions, including planes.

Yes, the equation x+2y+3z=0 can be solved for x, y, and z. However, since it is a linear equation with three variables, there are infinitely many solutions. To solve for a specific value of x, y, or z, you would need additional information or constraints.

The language of mathematics is particularly effective in representing relationshipsbetween two or more variables. As an example, let us consider the distance traveledin a certain length of time by a car moving at a constant speed of 40 miles per hour.We can represent this relationship by

The pair of numbers 1 and 40, considered together, is called a solution of theequation d = 40r because when we substitute 1 for t and 40 for d in the equation,we get a true statement. If we agree to refer to the paired numbers in a specifiedorder in which the first number refers to time and the second number refers todistance, we can abbreviate the above solutions as (1, 40), (2, 80), (3, 120), andso on. We call such pairs of numbers ordered pairs, and we refer to the first andsecond numbers in the pairs as components. With this agreement, solutions of theequation d - 40t are ordered pairs (t, d) whose components satisfy the equation.Some ordered pairs for t equal to 0, 1, 2, 3, 4, and 5 are

In any particular equation involving two variables, when we assign a value to oneof the variables, the value for the other variable is determined and thereforedependent on the first. It is convenient to speak of the variable associated with thefirst component of an ordered pair as the independent variable and the variableassociated with the second component of an ordered pair as the dependent variable. If the variables x and y are used in an equation, it is understood that replace-ments for x are first components and hence x is the independent variable andreplacements for y are second components and 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 termsof x. It is often easier to obtain solutions if equations are first expressed in such formbecause the dependent variable is expressed explicitly in terms of the independentvariable.

We obtained Equation (2) by adding the same quantity, -2x, to each memberof Equation (1), in that way getting y by itself. In general, we can write equivalentequations in two variables by using the properties we introduced in Chapter 3,where we solved first-degree equations in one variable.

Sometimes, we use a special notation to name the second component of an orderedpair that is paired with a specified first component. The symbol f(x), which is oftenused to name an algebraic expression in the variable x, can also be used to denotethe value of the expression for specific values of x. For example, if

In Section 1.1, we saw that every number corresponds to a point in a line. Simi-larly, every ordered pair of numbers (x, y) corresponds to a point in a plane. Tograph an ordered pair of numbers, we begin by constructing a pair of perpendicularnumber lines, called axes. The horizontal axis is called the x-axis, the vertical axisis called the y-axis, and their point of intersection is called the origin. These axesdivide the plane into four quadrants, as shown in Figure 7.1.

Now we can assign an ordered pair of numbers to a point in the plane by referringto the perpendicular distance of the point from each of the axes. If the firstcomponent is positive, the point lies to the right of the vertical axis; if negative, itlies to the left. If the second component is positive, the point lies above thehorizontal axis; if negative, it lies below.

The distance y that the point is located from the x-axis is called the ordinateof the point, and the distance x that the point is located from the y-axis is calledthe abscissa of the point. The abscissa and ordinate together are called the rectan-gular or Cartesian coordinates of the point (see Figure 7.2).

In Section 7.1, we saw that a solution of an equation in two variables is an orderedpair. In Section 7.2, we saw that the components of an ordered pair are thecoordinates of a point in a plane. Thus, to graph an equation in two variables, wegraph the set of ordered pairs that are solutions to the equation. For example, wecan find some solutions to the first-degree equation

In the above example, the values we used forx were chosen at random; we could have usedany values of x to find solutions to the equation.The graphs of any other ordered pairs that are solutions of the equation would alsobe on the line shown in Figure 7.3. In fact, each linear equation in two variableshas an infinite number of solutions whose graph lies on a line. However, we onlyneed to find two solutions because only two points are necessary to determine astraight 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 byassigning any convenient value to one variable and determining the corre-sponding value of the other variable.Graph these ordered pairs.Draw a straight line through the points.Check by graphing a third ordered pair that is a solution of the equation andverify that it lies on the line.

In Section 7.3, we assigned values to x in equations in two variables to find thecorresponding values of y. The solutions of an equation in two variables that aregenerally easiest to find are those in which either the first or second component is0. For example, if we substitute 0 for x in the equation

Thus, a solution of Equation (1) is (0, 3). We can also find ordered pairs that aresolutions of equations in two variables by assigning values to y and determining thecorresponding values of x. In particular, if we substitute 0 for y in Equation (1), weget

This method of drawing the graph of a linear equation is called the interceptmethod of graphing. Note that when we use this method of graphing a linearequation, there is no advantage in first expressing y explicitly in terms of x.

The ordered pairs (3, 0) and (0, -6) are solutions of 2x - y = 6. Graphing thesepoints and connecting them with a straight line give us the graph of 2x - y = 6.If the graph intersects the axes at or near the origin, the intercept method is notsatisfactory. We must then graph an ordered pair that is a solution of the equationand whose graph is not the origin or is not too close to the origin.

It is often convenient to use a special notation to distinguish between the rectan-gular coordinates of two different points. We can designate one pair of coordinatesby (x1, y1 (read "x sub one, y sub one"), associated with a point P1, and a secondpair of coordinates by (x2, y2), associated with a second point P2, as shown in Figure7.7. Note in Figure 7.7 that when going from P1 to P2, the vertical change (orvertical distance) between the two points is y2 - y1 and the horizontal change (orhorizontal distance) is x2 - x1.

Lines with various slopes are shown in Figure 7.8 below. Slopes of the lines thatgo up to the right are positive (Figure 7.8a) and the slopes of lines that go downto the right are negative (Figure 7.8b). And note (Figure 7.8c) that because allpoints on a horizontal line have the same y value, y2 - y1 equals zero for any twopoints and the slope of the line is simply

Equation (2) is called the point-slope form for a linear equation. In Equation (2),m, x1 and y1 are known and x and y are variables that represent the coordinates ofany point on the line. Thus, whenever we know the slope of a line and a point onthe line, we can find the equation of the line by using Equation (2).

In a direct variation, if we know a set of conditions on the two variables, and ifwe further know another value for one of the variables, we can find the value ofthe second variable for this new set of conditions.

In general, to graph a first-degree inequality in two variables of the formAx + By = C or Ax + By = C, we first graph the equation Ax + By = C andthen determine which half-plane (a region above or below the line) contains thesolutions. We then shade this half-plane. We can always determine which half-plane to shade by selecting a point (not on the line of the equation Ax + By = C)and testing to see if the ordered pair associated with the point is a solution of thegiven inequality. 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.

Solution

We begin by graphing the line y = 2x (see graph a). Since the line passes throughthe origin, we must choose another point not on the line as our test point. We willuse (0, 1). Since the statement

A solution of an equation in two variables is an ordered pair of numbers. In theordered pair (x, y), x is called the first component and y is called the secondcomponent. For an equation in two variables, the variable associated with the firstcomponent of a solution is called the independent variable and the variableassociated with the second component is called the dependent variable.Function notation f(x) is used to name an algebraic expression in x. When x inthe symbol f(x) is replaced by a particular value, the symbol represents the valueof the expression for that value of x.

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