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Chapter 3: Problem 9

Clear the fractions, and rewrite each equation in standard form. \(y-3=\frac{5}{8}(x+1)\)

Short Answer

Expert verified
5x - 8y = -29

Step by step solution

01

Eliminate the fraction

To clear the fraction, multiply each term in the equation by 8, the denominator of the fraction: y - 3 = \frac{5}{8}(x + 1) \[8 \times (y - 3) = 8 \times \frac{5}{8}(x + 1)\] Simplify to get: \[8y - 24 = 5(x + 1)\]
02

Distribute and simplify

Distribute 5 on the right side of the equation: \[8y - 24 = 5x + 5\]
03

Rewrite in standard form

Rearrange the equation to be in standard form, which is Ax + By = C. Move all terms to one side and simplify: \[8y - 24 - 5x - 5 = 0\] Combine like terms: \[ -5x + 8y - 29 = 0\] Rearrange to standard form if needed: \[ 5x - 8y = -29\]

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Key Concepts

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

Standard Form Equations
Standard form equations in algebra are expressed as Ax + By = C, where A, B, and C are integers. This format is neat and consistent, making it easier to compare and analyze different equations. To get an equation into standard form, you often need to rearrange and simplify terms. For example, for the equation 5x + 8y = -29, A is 5, B is -8, and C is -29.
Notice how getting rid of fractions and distributing terms make it easier to reach the standard form.
Distributive Property
The distributive property allows you to multiply a value outside a set of parentheses by each term inside the parentheses. This helps in breaking down complex algebraic expressions. For instance, in the equation 8y - 24 = 5(x + 1), we apply the distributive property to 5(x + 1), resulting in 5x + 5.
Using this property simplifies expressions, making them easier to handle and solve.
Simplifying Algebraic Expressions
Simplifying algebraic expressions involves combining like terms and performing operations to make the expression easier to read and solve. In our exercise, 8y - 24 = 5x + 5, combine and simplify to remove any fractions or complex terms. Always check to ensure every term is simplified.
This step is crucial in getting your equation ready for further operations like rearranging or solving.
Rearranging Equations
Rearranging equations involves moving terms from one side of the equal sign to the other to achieve a desired form, like the standard form. In the equation -5x + 8y - 29 = 0, you can move all terms to one side to isolate variables or constants. This helps in solving for specific variables and provides a structured format for analysis.
Getting comfortable with rearranging terms is key to mastering algebraic manipulations.

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Most popular questions from this chapter

(a) find the y-intercept. (b) find the x-intercept. (c) find a third solution of the equation. (d) graph the equation. \(3 x+2 y=4\)

(a) find three solutions of the equation. (b) graph the equation. \(y=-x\)

(a) graph the given points, and draw a line through the points. (b) use the graph to find the slope of the line. (c) use the slope formula to find the slope of the line. \((0,-7) ;(3,0)\)

If \(f(x)=8 x-9\), evaluate \(f\left(\frac{3}{5}\right)\).

Line \(A\) passes through \((4,10)\) and \((6,14)\). a. Find the slope of line \(A\). b. Line \(B\) is perpendicular to line \(A\). Identify the slope of line \(B\). c. Line \(B\) passes through the point \((-1,5)\). Write the equation of line \(B\) in slope-intercept form.
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