Rewrite The Given Equations So They Are In Slope-intercept Form ( Y = M X + B (y = Mx + B ( Y = M X + B ].a. 7 X − 14 Y = − 56 7x - 14y = -56 7 X − 14 Y = − 56 B. − 8 X − 2 Y = 6 -8x - 2y = 6 − 8 X − 2 Y = 6 C. 15 X + 9 Y = 45 15x + 9y = 45 15 X + 9 Y = 45

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 represented as $y = mx + b$, where $m$ is the slope of the line and $b$ is the y-intercept. In this article, we will explore how to rewrite the given equations in slope-intercept form.

Rewriting the First Equation

The first equation is $7x - 14y = -56$. To rewrite this equation in slope-intercept form, we need to isolate the variable $y$.

Step 1: Add 14y to both sides of the equation

$7x - 14y + 14y = -56 + 14y$

This simplifies to:

$7x = -56 + 14y$

Step 2: Subtract 7x from both sides of the equation

$7x - 7x = -56 + 14y - 7x$

This simplifies to:

$0 = -56 + 14y - 7x$

Step 3: Add 7x to both sides of the equation

$7x = -56 + 14y - 7x + 7x$

This simplifies to:

$7x = -56 + 14y$

Step 4: Subtract 7x from both sides of the equation

$7x - 7x = -56 + 14y - 7x$

This simplifies to:

$0 = -56 + 14y - 7x$

Step 5: Add 56 to both sides of the equation

$0 + 56 = -56 + 56 + 14y - 7x$

This simplifies to:

$56 = 14y - 7x$

Step 6: Add 7x to both sides of the equation

$56 + 7x = 14y - 7x + 7x$

This simplifies to:

$56 + 7x = 14y$

Step 7: Subtract 7x from both sides of the equation

$56 + 7x - 7x = 14y - 7x + 7x$

This simplifies to:

$56 = 14y$

Step 8: Divide both sides of the equation by 14

$\frac{56}{14} = \frac{14y}{14}$

This simplifies to:

$4 = y$

Step 9: Rewrite the equation in slope-intercept form

$y = 4$

The final answer is $\boxed{y = 4}$.

Rewriting the Second Equation

The second equation is $-8x - 2y = 6$. To rewrite this equation in slope-intercept form, we need to isolate the variable $y$.

Step 1: Add 2y to both sides of the equation

$-8x - 2y + 2y = 6 + 2y$

This simplifies to:

$-8x = 6 + 2y$

Step 2: Subtract 6 from both sides of the equation

$-8x - 6 = 6 + 2y - 6$

This simplifies to:

$-8x - 6 = 2y$

Step 3: Add 6 to both sides of the equation

$-8x - 6 + 6 = 2y + 6$

This simplifies to:

$-8x = 2y + 6$

Step 4: Subtract 2y from both sides of the equation

$-8x - 2y = 2y + 6 - 2y$

This simplifies to:

$-8x - 2y = 6$

Step 5: Add 2y to both sides of the equation

$-8x - 2y + 2y = 6 + 2y$

This simplifies to:

$-8x = 6 + 2y$

Step 6: Subtract 6 from both sides of the equation

$-8x - 6 = 6 + 2y - 6$

This simplifies to:

$-8x - 6 = 2y$

Step 7: Add 6 to both sides of the equation

$-8x - 6 + 6 = 2y + 6$

This simplifies to:

$-8x = 2y + 6$

Step 8: Subtract 2y from both sides of the equation

$-8x - 2y = 2y + 6 - 2y$

This simplifies to:

$-8x - 2y = 6$

Step 9: Divide both sides of the equation by -2

$\frac{-8x}{-2} = \frac{6}{-2}$

This simplifies to:

$4x = -3$

Step 10: Divide both sides of the equation by 4

$\frac{4x}{4} = \frac{-3}{4}$

This simplifies to:

$x = -\frac{3}{4}$

Step 11: Rewrite the equation in slope-intercept form

$y = -4x - 3$

The final answer is $\boxed{y = -4x - 3}$.

Rewriting the Third Equation

The third equation is $15x + 9y = 45$. To rewrite this equation in slope-intercept form, we need to isolate the variable $y$.

Step 1: Subtract 15x from both sides of the equation

$15x + 9y - 15x = 45 - 15x$

This simplifies to:

$9y = 45 - 15x$

Step 2: Subtract 45 from both sides of the equation

$9y - 45 = 45 - 15x - 45$

This simplifies to:

$9y - 45 = -15x$

Step 3: Add 45 to both sides of the equation

$9y - 45 + 45 = -15x + 45$

This simplifies to:

$9y = -15x + 45$

Step 4: Subtract 45 from both sides of the equation

$9y - 45 = -15x + 45 - 45$

This simplifies to:

$9y - 45 = -15x$

Step 5: Add 45 to both sides of the equation

$9y - 45 + 45 = -15x + 45$

This simplifies to:

$9y = -15x + 45$

Step 6: Divide both sides of the equation by 9

$\frac{9y}{9} = \frac{-15x + 45}{9}$

This simplifies to:

$y = -\frac{15}{9}x + 5$

Step 7: Simplify the fraction

$y = -\frac{5}{3}x + 5$

The final answer is $\boxed{y = -\frac{5}{3}x + 5}$.

In the previous article, we explored how to rewrite equations in slope-intercept form. In this article, we will answer some common questions related to 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 represented as $y = mx + b$, 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 $y$ by performing a series of algebraic operations. The steps involved in rewriting an equation in slope-intercept form are:

1. Add or subtract the same value to both sides of the equation to isolate the term containing $y$.
2. Divide both sides of the equation by the coefficient of $y$ to solve for $y$.
3. Simplify the resulting expression to obtain the slope-intercept form of the equation.

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

A: The slope-intercept form and the standard form of a linear equation are two different ways of representing the same equation. The slope-intercept form is represented as $y = mx + b$, while the standard form is represented as $Ax + By = C$. The main difference between the two forms is that the slope-intercept form isolates the variable $y$, while the standard form isolates the variables $x$ and $y$.

Q: How do I determine the slope and y-intercept of a linear equation?

A: To determine the slope and y-intercept of a linear equation, you need to rewrite the equation in slope-intercept form. The slope is the coefficient of the $x$ term, while the y-intercept is the constant term.

Q: Can I rewrite an equation in slope-intercept form if it is not in the standard form?

A: Yes, you can rewrite an equation in slope-intercept form even if it is not in the standard form. However, you may need to perform additional algebraic operations to isolate the variable $y$.

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 $y$ correctly
• Not simplifying the resulting expression
• Not checking the equation for errors
• Not using the correct algebraic operations to isolate the variable $y$

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 value for $x$ and solve for $y$ to see if the equation is true
• Graph the equation to see if it is a straight line
• Use a calculator to check the equation
• Check the equation for errors

Q: Can I use a calculator to rewrite an equation in slope-intercept form?

A: Yes, you can use a calculator to rewrite an equation in slope-intercept form. However, you should always check your work to ensure that the equation is correct.

Q: How do I rewrite an equation in slope-intercept form if it has multiple variables?

A: To rewrite an equation in slope-intercept form if it has multiple variables, you need to isolate the variable $y$ by performing a series of algebraic operations. The steps involved in rewriting an equation in slope-intercept form with multiple variables are:

1. Isolate the term containing $y$ by adding or subtracting the same value to both sides of the equation.
2. Divide both sides of the equation by the coefficient of $y$ to solve for $y$.
3. Simplify the resulting expression to obtain the slope-intercept form of the equation.

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 if it has a fraction. However, you may need to perform additional algebraic operations to isolate the variable $y$.

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

A: To rewrite an equation in slope-intercept form if it has a negative exponent, you need to isolate the variable $y$ by performing a series of algebraic operations. The steps involved in rewriting an equation in slope-intercept form with a negative exponent are:

1. Isolate the term containing $y$ by adding or subtracting the same value to both sides of the equation.
2. Divide both sides of the equation by the coefficient of $y$ to solve for $y$.
3. Simplify the resulting expression to obtain the slope-intercept form of the equation.

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

A: Yes, you can rewrite an equation in slope-intercept form if it has a radical. However, you may need to perform additional algebraic operations to isolate the variable $y$.

Q: How do I rewrite an equation in slope-intercept form if it has a trigonometric function?

A: To rewrite an equation in slope-intercept form if it has a trigonometric function, you need to isolate the variable $y$ by performing a series of algebraic operations. The steps involved in rewriting an equation in slope-intercept form with a trigonometric function are:

1. Isolate the term containing $y$ by adding or subtracting the same value to both sides of the equation.
2. Divide both sides of the equation by the coefficient of $y$ to solve for $y$.
3. Simplify the resulting expression to obtain the slope-intercept form of the equation.

In conclusion, rewriting equations in slope-intercept form is a crucial skill in mathematics that helps us understand the relationship between the variables in an equation. By following the steps outlined in this article, we can rewrite the given equations in slope-intercept form and gain a deeper understanding of the underlying mathematics.