Rewrite The Following Equation In Slope-intercept Form: 18 X − 5 Y = − 2 18x - 5y = -2 18 X − 5 Y = − 2 Write Your Answer Using Integers, Proper Fractions, And Improper Fractions In Simplest Form. □ \square □

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Introduction

In mathematics, the slope-intercept form of a linear equation is a fundamental concept that helps us understand the relationship between the variables in an equation. The slope-intercept form is given by the equation y = mx + b, where m is the slope of the line and b is the y-intercept. In this article, we will rewrite the given equation in slope-intercept form and provide a step-by-step solution.

The Given Equation

The given equation is 18x−5y=−218x - 5y = -2. Our goal is to rewrite this equation in slope-intercept form, which is y = mx + b.

Step 1: Isolate the Variable y

To rewrite the equation in slope-intercept form, we need to isolate the variable y. We can do this by adding 5y to both sides of the equation, which gives us:

18x−5y+5y=−2+5y18x - 5y + 5y = -2 + 5y

This simplifies to:

18x=−2+5y18x = -2 + 5y

Step 2: Add 2 to Both Sides

Next, we add 2 to both sides of the equation to get:

18x+2=5y18x + 2 = 5y

Step 3: Divide Both Sides by 5

Now, we divide both sides of the equation by 5 to isolate y:

18x+25=y\frac{18x + 2}{5} = y

Step 4: Simplify the Equation

We can simplify the equation by combining the terms on the left-hand side:

18x5+25=y\frac{18x}{5} + \frac{2}{5} = y

Step 5: Write the Equation in Slope-Intercept Form

Finally, we can write the equation in slope-intercept form by rearranging the terms:

y=185x+25y = \frac{18}{5}x + \frac{2}{5}

Conclusion

In this article, we rewrote the given equation in slope-intercept form using a step-by-step approach. We isolated the variable y, added 2 to both sides, divided both sides by 5, and simplified the equation to obtain the final result. The equation in slope-intercept form is y=185x+25y = \frac{18}{5}x + \frac{2}{5}.

Key Takeaways

  • The slope-intercept form of a linear equation is given by y = mx + b.
  • To rewrite an equation in slope-intercept form, we need to isolate the variable y.
  • We can add or subtract terms from both sides of the equation to isolate y.
  • We can divide both sides of the equation by a non-zero constant to isolate y.
  • The final result should be in the form y = mx + b, where m is the slope and b is the y-intercept.

Examples and Applications

The slope-intercept form of a linear equation has many applications in mathematics and real-world problems. For example, it can be used to:

  • Find the equation of a line given its slope and y-intercept.
  • Determine the slope and y-intercept of a line given its equation.
  • Solve systems of linear equations.
  • Model real-world problems using linear equations.

Glossary of Terms

  • Slope-intercept form: The form of a linear equation given by y = mx + b, where m is the slope and b is the y-intercept.
  • Slope: The coefficient of the x-term in the slope-intercept form of a linear equation.
  • Y-intercept: The constant term in the slope-intercept form of a linear equation.
  • Linear equation: An equation of the form ax + by = c, where a, b, and c are constants.

References

  • [1] "Algebra" by Michael Artin.
  • [2] "Linear Algebra and Its Applications" by Gilbert Strang.
  • [3] "Mathematics for Computer Science" by Eric Lehman, F Thomson Leighton, and Albert R Meyer.
    Frequently Asked Questions (FAQs) about Rewriting Equations in Slope-Intercept Form =====================================================================================

Q: What is the slope-intercept form of a linear equation?

A: The slope-intercept form of a linear equation is given by y = mx + b, where m is the slope of the line and b is the y-intercept.

Q: How do I rewrite an equation in slope-intercept form?

A: To rewrite an equation in slope-intercept form, you need to isolate the variable y. You can do this by adding or subtracting terms from both sides of the equation, and then dividing both sides by a non-zero constant.

Q: What is the slope of a line in slope-intercept form?

A: The slope of a line in slope-intercept form is the coefficient of the x-term, which is m.

Q: What is the y-intercept of a line in slope-intercept form?

A: The y-intercept of a line in slope-intercept form is the constant term, which is b.

Q: Can I rewrite an equation in slope-intercept form if it has a fraction?

A: Yes, you can rewrite an equation in slope-intercept form even if it has a fraction. You just need to simplify the fraction and then isolate the variable y.

Q: How do I simplify a fraction in slope-intercept form?

A: To simplify a fraction in slope-intercept form, you can multiply both the numerator and the denominator by the same non-zero constant.

Q: Can I rewrite an equation in slope-intercept form if it has a negative slope?

A: Yes, you can rewrite an equation in slope-intercept form even if it has a negative slope. The slope-intercept form of a linear equation with a negative slope is given by y = -mx + b.

Q: What is the difference between the slope-intercept form and the standard form of a linear equation?

A: The slope-intercept form of a linear equation is given by y = mx + b, while the standard form is given by ax + by = c. The slope-intercept form is more convenient for graphing and solving problems, while the standard form is more convenient for solving systems of linear equations.

Q: Can I use the slope-intercept form to solve a system of linear equations?

A: Yes, you can use the slope-intercept form to solve a system of linear equations. You can rewrite each equation in slope-intercept form and then solve for the variables.

Q: What are some common mistakes to avoid when rewriting equations in slope-intercept form?

A: Some common mistakes to avoid when rewriting equations in slope-intercept form include:

  • Not isolating the variable y
  • Not simplifying fractions
  • Not checking for negative slopes
  • Not using the correct form of the equation

Q: How do I check my work when rewriting an equation in slope-intercept form?

A: To check your work when rewriting an equation in slope-intercept form, you can:

  • Plug in a point on the line to see if it satisfies the equation
  • Graph the line and check if it passes through the point
  • Use a calculator to check if the equation is true

Q: What are some real-world applications of rewriting equations in slope-intercept form?

A: Some real-world applications of rewriting equations in slope-intercept form include:

  • Modeling population growth
  • Calculating the cost of goods
  • Determining the slope of a roof
  • Finding the equation of a line given its slope and y-intercept

Q: Can I use rewriting equations in slope-intercept form to solve problems in other areas of mathematics?

A: Yes, you can use rewriting equations in slope-intercept form to solve problems in other areas of mathematics, such as:

  • Algebra
  • Geometry
  • Trigonometry
  • Calculus

Q: What are some tips for mastering the skill of rewriting equations in slope-intercept form?

A: Some tips for mastering the skill of rewriting equations in slope-intercept form include:

  • Practicing regularly
  • Using online resources and tutorials
  • Working with a tutor or teacher
  • Breaking down complex problems into simpler ones

Q: How long does it take to master the skill of rewriting equations in slope-intercept form?

A: The amount of time it takes to master the skill of rewriting equations in slope-intercept form depends on the individual and their level of math proficiency. With regular practice and dedication, it can take anywhere from a few weeks to a few months to become proficient in rewriting equations in slope-intercept form.