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Chapter 7: Problem 1

The following equations all represent the same line. Which one is in standard form as specified in this section? A. \(3 x-2 y=5\) B. \(y=\frac{3}{2} x-\frac{5}{2}\) C. \(\frac{3}{5} x-\frac{2}{5} y=1\) D. \(3 x=2 y+5\)

Short Answer

Expert verified
A. \[3x - 2y = 5\] is in standard form.

Step by step solution

01

Identify Standard Form

The standard form of a linear equation is given by the format: \[Ax + By = C\], where A, B, and C are integers, and A should be a positive integer.
02

Check Equation A

Equation A is given as \[3x - 2y = 5\]. This fits the format \[Ax + By = C\] with \(A=3\), \(B=-2\), and \(C=5\). Equation A is in standard form.
03

Check Equation B

Equation B is given as \[y = \frac{3}{2}x - \frac{5}{2}\]. This is in slope-intercept form and needs to be converted:\[\frac{3}{2}x - y = \frac{5}{2}\]. Multiply all terms by 2 to clear the fractions:\[3x - 2y = 5\]. This matches Equation A, which is in standard form.
04

Check Equation C

Equation C is given as \[\frac{3}{5}x - \frac{2}{5}y = 1\]. To convert to standard form, multiply all terms by 5 to clear the fractions:\[3x - 2y = 5\]. This matches Equation A, which is in standard form.
05

Check Equation D

Equation D is given as \[3x = 2y + 5\]. To convert it to standard form, rearrange the equation:\[3x - 2y = 5\]. This matches Equation A, which is in standard form.
06

Conclusion

All equations can be converted to the form \[3x - 2y = 5\], which is in standard form (\[Ax + By = C\]) according to the specified section.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Standard Form
The standard form of a linear equation is an important way to describe a line. It's written as Ax + By = C , where A, B, and C are integers, and A should be a positive integer. This form is very useful when you need to find both the x and y intercepts easily. You'll notice that a common feature is that 'x' and 'y' are both on the left side of the equation, while the constant is on the right side.
Linear Equations
A linear equation is any equation that can be written in the form Ax + By = C. This form creates a straight line when plotted on a graph. Linear equations are fundamental in algebra and help us understand relationships between variables. They have no exponents or powers and always represent a straight line. Here are some key points:

  • The graph forms a straight line.
  • The equation has no variables with exponents other than 1.
  • Linear equations in standard form usually help in identifying x and y intercepts quickly.
Slope-Intercept Form
The slope-intercept form is another way to write linear equations. It's given by the format y = mx + b where 'm' is the slope and 'b' is the y-intercept. This form is particularly useful for quickly identifying the slope and y-intercept of the line. For example: y = (3/2)x - (5/2)
  • 'm' represents how steep the line is. For instance, if m=3/2, for every 2 units you move right, you move 3 units up.
  • 'b' tells you where the line crosses the y-axis. For example, when b=-5/2, the line crosses the y-axis at -2.5.
You can convert from slope-intercept form to standard form with basic algebra, such as moving all terms involving variables to one side and clearing fractions.
Equation Conversion
Converting equations between forms is a useful algebra skill. It helps to rewrite them based on what makes solving easier or what the situation requires. For example, if you start with slope-intercept form y = (3/2)x - (5/2) , you can convert it to standard form by moving terms around and getting rid of fractions.

Here's a step-by-step on how to do it:
  • Start with y = (3/2)x - (5/2)
  • Subtract (3/2)x from both sides to get - (3/2)x + y = - (5/2)
  • Multiply through by 2 to clear the fraction: -3x + 2y = -5
  • Multiply by -1 (if needed) to make A positive: 3x - 2y = 5
  • You've now converted the equation to standard form.

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Most popular questions from this chapter

Without graphing, do the following for each system of equations. (a) Describe each system. (b) State the number of solutions. (c) Is the system inconsistent, are the equations dependent, or neither? \(3 x+2 y=5\) \(6 x+4 y=3\)

Determine whether each pair of lines is parallel, perpendicular, or neither $$ 4 x-3 y=8 \text { and } 4 y+3 x=12 $$

Solve each system by graphing. If the system is inconsistent or the equations are dependent, say so. \(2 x-3 y=-6\) \(y=-3 x+2\)

A fruit drink is made by mixing juices. Such a drink with \(50 \%\) juice is to be mixed with a drink that is \(30 \%\) juice to obtain \(200 \mathrm{~L}\) of a drink that is \(45 \%\) juice. How much of each should be used?

Solve each system by graphing. If the system is inconsistent or the equations are dependent, say so. \(2 x-y=6\) \(4 x-2 y=8\)
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