Solving Equations and Graphing
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1 Solving Equations and Graphing Question 1: How do you solve a linear equation? Answer 1: 1. Remove any parentheses or other grouping symbols (if necessary). 2. If the equation contains a fraction, multiply through by the least common denominator to clear out the denominators (if necessary). 3. Add or subtract terms to both sides of the equation to get all of the variables onto one side of the equation and all the constants onto the other side. 4. Divide both sides of the equation by the coefficient (number in front of the variable). 5. Simplify the solution (if necessary). 6. Check your solution by plugging it into the original equation to see if it makes the equation true. Question 2: What is an example of an equation that contains both parentheses and fractions? Answer 2: First, the parentheses are removed by using the distributive property. Multiply through by the least common denominator (LCD). The only denominator is 2, so multiply each term by 2 and simplify. (x + 2) = 8x 2x + 6 x + 2 = 6x + 6 Then, like terms are collected on each side of the equation. Get all x terms on one side and all constants on the other using addition or subtraction. Divide both sides by 5. x = 6x x 4 = 6x x x 4 = 6x x -4 = 5x x = -4/5 or -0.8
2 Remember to check x = -4/5 into the original equation to verify that it makes the equation true. It is easy to make a mistake when solving the equation, so be sure to check your answer. Question 3: What is an example of the differences between solving a linear equation and a linear inequality? Answer 3: Notice the similarities and differences. First, subtract 1 from both sides. -2x = 5 Divide by -2 to get x on a side by itself. When it is divided by -2, the direction of the inequality must change. Only when you multiply or divide through by a negative number does the direction of the inequality change. x = -5 / 2 Solve -2x + 1 > 6. First, subtract 1 from both sides. -2x > 5 Divide by -2 to get x on a side by itself. When it is divided by -2, the direction of the inequality must change. Only when you multiply or divide through by a negative number does the direction of the inequality change. x < -5 / 2 Question 4: How are equations for linear application problems set up? Answer 4: One classic linear application is the perimeter problem. Suppose that the length of a rectangle is 6 cm longer than the width and that the perimeter is 18.
3 You want to write the equation with only one variable. The perimeter is the distance around the rectangle. Another example of how to set up a linear equation is as follows: Suppose that the total number of accounting, business, and art students in a class is 13. The number of art majors in that class is 2 times the number of business majors, and the number of accounting majors is 3 less than the number of business majors. Both the accounting majors and art majors are given in terms of the number of business majors, so let x equal the number of business majors. The number of art majors = 2x (twice the number of business students). The number of accounting majors = x 3 (3 less than the number of business students). The number of business students + the number of art students + the number of accounting students = 13. Therefore, the equation would be x + 2x + x - 3 = 13. Question 5: How do you solve formulas? Answer 5: Formulas are often used in everyday life, for example, when finding the volume of a can or the interest on a loan. Linear equations contain one variable, such as 2x + 3 = 5, but a formula usually contains more than one variable. An example is the distance formula, d = rt. When solving these formulas, the idea is to treat all the variables except the one that you are solving for as constants. Get the variable that you are solving for on a side by itself. First subtract 2x from both sides. 2x - 2x + 3y = 21-2x 3y = 21 2x Next divide both sides by 3. 3y / 3 = 21 / 3 2x / 3
4 y = 7 2x / 3 Usually the x term is written first, so rearrange the right side. or Question 6: What is an example of a linear inequality application problem? Answer 6: A bank offers two checking account plans. Plan A has a base service charge of $4 a month plus $0.10 per check. Plan B charges a base service charge of $3 a month plus $0.15 per check. Let x be the number of checks written in a month. By how many checks per month will plan A be better than plan B? Cost of plan A = x Cost of plan B = x Cost of plan A < Cost of plan B x < x -.05x < -1 x > 20 checks Question 7: What is the rectangular coordinate system? Answer 7: The rectangular coordinate system is used to graph points and equations. To create the rectangular coordinate system, use the following steps: 1. Draw a horizontal line. 2. Choose a point on the line. This point is called the origin and is assigned a value of Choose a length, called the scale, and mark it off the right of the origin. Continue writing the measurements to create the positive integers. 4. Complete the same process on the left side of the origin. It is a reflection except these are all the negatives of the real numbers. 5. Draw a vertical line passing through the origin. 6. Label the real numbers the same as the horizontal line with the numbers above the horizontal line as positive and the ones below as negative.
5 The horizontal line is usually called the x-axis and the vertical line is usually called the y-axis. These are arbitrary names and can be called by any name as the application may need, like hours, miles, or area. The scaling can also be changed according to the need. Points can be plotted on the coordinate system by using ordered pairs known as coordinates. These points are denoted by using parenthesis. For example, in the point (2, 3), 2 represents the x-value and 3 represents the y-value. To plot the points, locate 2 on the x-axis, then go vertically upward 3 units.
6 Question 8: What are the various forms of a line? Answer 8: The standard form of an equation of a line is ax + by + c = 0. This form shows that the function is a line but does not give other information about the line. The standard form can be converted to the slope-intercept form. Every nonvertical line can be written in the slope-intercept form. This form is y = mx + b where m is the slope of the line and b is the y-intercept. The y-intercept is the point where the line crosses the y-axis. The slope is the steepness of the line and is calculated by the rise divided by the run. Question 9: How do you write an equation of a line? Answer 9: Given the slope of a line and the y-intercept, the equation of the line can be directly written. For example, if the slope m = 2 and the y-intercept is 3, then the equation of the line is y = 2x + 3. When given two points, the first step is to find the slope. To calculate the slope, divide the
7 difference in y-values by the difference in x-values. For example, if the line passes through the two points (1,1) and (3,5), then the slope is (5-1) / (3-1) = 4 / 2 = 2. To find b, use the calculated slope and either point to substitue in the slope-intercept form. The slope-intercept form would be y = 2x + b, and substitue (1, 1) into the equation. This gives 1 = 2(1) b or b = -1. Input the information into slope-intercept form where m = 2 and b = -1, and the equation of the line is y = 2x - 1.
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