Systems of Inequalities Related Pages We first need to review the symbols for inequalities: There are endless solutions for inequalities.
We will assign a number to a line, which we call slope, that will give us a measure of the "steepness" or "direction" of the line. 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 coordinates by x1, y1 read "x sub one, y sub one"associated with a point P1, and a second pair of coordinates by x2, y2associated with a second point P2, as shown in Figure 7. Note in Figure 7.
The ratio of the vertical change to the horizontal change is called the slope of the line containing the points P1 and P2. This ratio is usually designated by m. Thus, Example 1 Find the slope of the line containing the two points with coordinates -4, 2 and 3, 5 as shown in the figure at the right.
Solution We designate 3, 5 as x2, y2 and -4, 2 as x1, y1. Substituting into Equation 1 yields Note that we get the same result if we subsitute -4 and 2 for x2 and y2 and 3 and 5 for x1 and y1 Lines with various slopes are shown in Figure 7.
Slopes of the lines that go up to the right are positive Figure 7. And note Figure 7. However, is undefined, so that a vertical line does not have a slope. In this case, These lines will never intersect and are called parallel lines. Now consider the lines shown in Figure 7. In this case, These lines meet to form a right angle and are called perpendicular lines.
In general, if two lines have slopes and m2: If we denote any other point on the line as P x, y See Figure 7. In general let us say we know a line passes through a point P1 x1, y1 and has slope m.
If we denote any other point on the line as P x, y see Figure 7.
In Equation 2m, x1 and y1 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 can find the equation of the line by using Equation 2.
Example 1 A line has slope -2 and passes through point 2, 4. Find the equation of the line. The slope and y-intercept can be obtained directly from an equation in this form. Example 2 If a line has the equation then the slope of the line must be -2 and the y-intercept must be 8.
Solution We first solve for y in terms of x by adding -2x to each member. We say that the variable y varies directly as x. Example 1 We know that the pressure P in a liquid varies directly as the depth d below the surface of the liquid.
In this section we will graph inequalities in two variables. That is, a, b is a solution of the inequality if the inequality is a true statement after we substitute a for x and b for y.
Thus, every point on or below the line is in the graph. We represent this 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 through the origin, we must choose another point not on the line as our test point.
We will use 0, 1. In the ordered pair x, yx is called the first component and y is called the second component. For an equation in two variables, the variable 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 to name an algebraic expression in x. When x in the symbol f x is replaced by a particular value, the symbol represents the 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.
The components of an ordered pair x, y associated with 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 straight line.Graphing Slope. Accurately graphing slope is the key to graphing linear equations. In the previous lesson, Calculating Slope, you learned how to calculate the slope of a line. In this lesson, you are going to graph a line, given the slope.
Home; Calculators; Algebra II Calculators; Math Problem Solver (all calculators) System of Linear Equations Calculator. This calculator will solve the system of linear equations of any kind, with steps shown, using either the Gauss-Jordan Elimination method or the Cramer's Rule.
Retail Catalog NoR SPRING err.^s^rr^_rr +^ea 0 q Helping you make things better This comput. Systems of linear inequalities A system of linear inequalities in two variables consists of at least two linear inequalities in the same variables. The solution of a linear inequality is the ordered pair that is a solution to all inequalities in the system and the graph of the linear inequality is .
Fit an algebraic two-variable inequality to its appropriate graph. Pearson Prentice Hall and our other respected imprints provide educational materials, technologies, assessments and related services across the secondary curriculum.