Convert The Following Equation Into Standard Form:$\[ Y = -\frac{7x}{8} + 4 \\]Rewrite In The Form \[$[?]x + \square Y = C\$\].

Introduction

In mathematics, linear equations are a fundamental concept that is used to describe the relationship between two or more variables. The standard form of a linear equation is a crucial aspect of algebra, as it allows us to easily identify the coefficients of the variables and the constant term. In this article, we will focus on converting the given equation into standard form, which is in the form of $ax + by = c$.

Understanding the Given Equation

The given equation is $y = -\frac{7x}{8} + 4$. This equation represents a linear relationship between the variables $x$ and $y$. The coefficient of $x$ is $-\frac{7}{8}$, and the constant term is $4$. To convert this equation into standard form, we need to isolate the variable $y$ on one side of the equation.

Converting the Equation to Standard Form

To convert the equation into standard form, we need to multiply both sides of the equation by the reciprocal of the coefficient of $x$. In this case, the reciprocal of $-\frac{7}{8}$ is $-\frac{8}{7}$. Multiplying both sides of the equation by $-\frac{8}{7}$, we get:

$$-\frac{8}{7} \cdot y = -\frac{8}{7} \cdot \left(-\frac{7x}{8} + 4\right)$$

Simplifying the right-hand side of the equation, we get:

$$-\frac{8}{7} \cdot y = x - \frac{32}{7}$$

Now, we can rewrite the equation in the form $ax + by = c$ by multiplying both sides of the equation by $-1$:

$$\frac{8}{7}y = -x + \frac{32}{7}$$

Multiplying both sides of the equation by $7$, we get:

$$8y = -7x + 32$$

Now, we can rewrite the equation in the standard form $ax + by = c$:

$$-7x + 8y = 32$$

Conclusion

In this article, we have successfully converted the given equation into standard form. The standard form of a linear equation is a crucial aspect of algebra, as it allows us to easily identify the coefficients of the variables and the constant term. By following the steps outlined in this article, we can convert any linear equation into standard form.

Key Takeaways

• The standard form of a linear equation is $ax + by = c$.
• To convert a linear equation into standard form, we need to isolate the variable $y$ on one side of the equation.
• We can use the reciprocal of the coefficient of $x$ to multiply both sides of the equation and convert it into standard form.
• The standard form of a linear equation allows us to easily identify the coefficients of the variables and the constant term.

Examples and Exercises

1. Convert the equation $y = 3x - 2$ into standard form.
2. Convert the equation $y = -2x + 1$ into standard form.
3. Convert the equation $y = \frac{1}{2}x + 3$ into standard form.

Solutions

1. To convert the equation $y = 3x - 2$ into standard form, we need to multiply both sides of the equation by $-1$:

$$-y = -3x + 2$$

Multiplying both sides of the equation by $-1$, we get:

$$y = 3x - 2$$

2. To convert the equation $y = -2x + 1$ into standard form, we need to multiply both sides of the equation by $-1$:

$$-y = 2x - 1$$

Multiplying both sides of the equation by $-1$, we get:

$$y = -2x + 1$$

3. To convert the equation $y = \frac{1}{2}x + 3$ into standard form, we need to multiply both sides of the equation by $-2$:

$$-2y = -x - 6$$

Multiplying both sides of the equation by $-1$, we get:

$$2y = x + 6$$

Conclusion

Introduction

In our previous article, we discussed how to convert a linear equation into standard form. In this article, we will provide a Q&A section to help you better understand the concept and address any questions you may have.

Q: What is the standard form of a linear equation?

A: The standard form of a linear equation is $ax + by = c$, where $a$ and $b$ are the coefficients of the variables $x$ and $y$, and $c$ is the constant term.

Q: How do I convert a linear equation into standard form?

A: To convert a linear equation into standard form, you need to isolate the variable $y$ on one side of the equation. You can do this by multiplying both sides of the equation by the reciprocal of the coefficient of $x$.

Q: What is the reciprocal of a coefficient?

A: The reciprocal of a coefficient is obtained by flipping the numerator and denominator of the fraction. For example, the reciprocal of $-\frac{7}{8}$ is $-\frac{8}{7}$.

Q: How do I multiply both sides of the equation by the reciprocal of the coefficient?

A: To multiply both sides of the equation by the reciprocal of the coefficient, you need to multiply both sides of the equation by the reciprocal of the coefficient and then simplify the right-hand side of the equation.

Q: What if the coefficient of $x$ is a fraction?

A: If the coefficient of $x$ is a fraction, you need to multiply both sides of the equation by the reciprocal of the fraction. For example, if the coefficient of $x$ is $-\frac{7}{8}$, you need to multiply both sides of the equation by $-\frac{8}{7}$.

Q: Can I convert a linear equation into standard form by multiplying both sides of the equation by a different number?

A: No, you cannot convert a linear equation into standard form by multiplying both sides of the equation by a different number. You need to multiply both sides of the equation by the reciprocal of the coefficient of $x$.

Q: What if the linear equation has a fraction as the constant term?

A: If the linear equation has a fraction as the constant term, you need to multiply both sides of the equation by the reciprocal of the fraction to eliminate the fraction.

Q: Can I convert a linear equation into standard form if it has a variable as the constant term?

A: No, you cannot convert a linear equation into standard form if it has a variable as the constant term. You need to isolate the variable $y$ on one side of the equation and then simplify the right-hand side of the equation.

Q: How do I know if a linear equation is in standard form?

A: You can check if a linear equation is in standard form by looking at the coefficients of the variables $x$ and $y$ and the constant term. If the coefficients of the variables $x$ and $y$ are integers and the constant term is a number, then the linear equation is in standard form.

Conclusion

In this Q&A article, we have addressed some common questions about converting linear equations to standard form. We hope that this article has helped you better understand the concept and has provided you with the information you need to convert linear equations to standard form.

Key Takeaways

• The standard form of a linear equation is $ax + by = c$.
• To convert a linear equation into standard form, you need to isolate the variable $y$ on one side of the equation.
• You can multiply both sides of the equation by the reciprocal of the coefficient of $x$ to convert it into standard form.
• If the coefficient of $x$ is a fraction, you need to multiply both sides of the equation by the reciprocal of the fraction.
• If the linear equation has a fraction as the constant term, you need to multiply both sides of the equation by the reciprocal of the fraction to eliminate the fraction.

Examples and Exercises

1. Convert the equation $y = 3x - 2$ into standard form.
2. Convert the equation $y = -2x + 1$ into standard form.
3. Convert the equation $y = \frac{1}{2}x + 3$ into standard form.

Solutions

1. To convert the equation $y = 3x - 2$ into standard form, we need to multiply both sides of the equation by $-1$:

$$-y = -3x + 2$$

Multiplying both sides of the equation by $-1$, we get:

$$y = 3x - 2$$

2. To convert the equation $y = -2x + 1$ into standard form, we need to multiply both sides of the equation by $-1$:

$$-y = 2x - 1$$

Multiplying both sides of the equation by $-1$, we get:

$$y = -2x + 1$$

3. To convert the equation $y = \frac{1}{2}x + 3$ into standard form, we need to multiply both sides of the equation by $-2$:

$$-2y = -x - 6$$

Multiplying both sides of the equation by $-1$, we get:

$$2y = x + 6$$