## Lesson 7 Homework Practice: How to Subtract Linear Expressions

Linear expressions are algebraic expressions that involve only constants and variables raised to the first power. For example, 3x - 5 and 2y + 7 are linear expressions, but x^2 - 4 and y / 2 are not.

## lesson 7 homework practice subtract linear expressions

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To subtract linear expressions, we need to combine the like terms. Like terms are terms that have the same variable and the same exponent. For example, 3x and -2x are like terms, but 3x and 2y are not.

To combine like terms, we add or subtract their coefficients. The coefficient is the number in front of the variable. For example, the coefficient of 3x is 3, and the coefficient of -2x is -2.

Let's look at an example of how to subtract linear expressions:

Example: Subtract (5x - 2) - (3x + 4).

Solution: To subtract two linear expressions, we need to distribute the negative sign to the second expression. This means we change the sign of each term in the second expression. So, we have:

(5x - 2) - (3x + 4) = (5x - 2) + (-3x - 4)

Next, we combine the like terms by adding or subtracting their coefficients. We have two like terms: 5x and -3x. To combine them, we add their coefficients: 5 + (-3) = 2. So, we have:

(5x - 2) + (-3x - 4) = 2x - 6

This is our final answer. We cannot simplify it further because there are no more like terms to combine.

Let's try another example of how to subtract linear expressions:

Example: Subtract (2x - 3y + 5) - (x + 2y - 7).

Solution: Again, we need to distribute the negative sign to the second expression. This means we change the sign of each term in the second expression. So, we have:

(2x - 3y + 5) - (x + 2y - 7) = (2x - 3y + 5) + (-x - 2y + 7)

Next, we combine the like terms by adding or subtracting their coefficients. We have three pairs of like terms: 2x and -x, -3y and -2y, and 5 and 7. To combine them, we add their coefficients: 2 + (-1) = 1, -3 + (-2) = -5, and 5 + 7 = 12. So, we have:

(2x - 3y + 5) + (-x - 2y + 7) = x - 5y + 12

This is our final answer. We cannot simplify it further because there are no more like terms to combine.

Sometimes, we may need to use the distributive property to simplify linear expressions before subtracting them. The distributive property states that a(b + c) = ab + ac. This means we can multiply a term outside a parenthesis by each term inside the parenthesis and add the results.

Let's look at an example of how to use the distributive property to subtract linear expressions:

Example: Subtract (3x - 4) - 2(4x + 1).

Solution: To subtract these linear expressions, we need to distribute the negative sign to the second expression. This means we change the sign of each term in the second expression. So, we have:

(3x - 4) - 2(4x + 1) = (3x - 4) + (-2)(4x + 1)

Next, we need to use the distributive property to multiply -2 by each term inside the parenthesis. This means we multiply -2 by 4x and by 1. So, we have:

(3x - 4) + (-2)(4x + 1) = (3x - 4) + (-8x - 2)

Next, we combine the like terms by adding or subtracting their coefficients. We have two pairs of like terms: 3x and -8x, and -4 and -2. To combine them, we add their coefficients: 3 + (-8) = -5, and -4 + (-2) = -6. So, we have:

(3x - 4) + (-8x - 2) = -5x - 6

This is our final answer. We cannot simplify it further because there are no more like terms to combine.

Let's try one more example of how to subtract linear expressions:

Example: Subtract (x - 2y + 3) - (2x - y - 4).

Solution: As before, we need to distribute the negative sign to the second expression. This means we change the sign of each term in the second expression. So, we have:

(x - 2y + 3) - (2x - y - 4) = (x - 2y + 3) + (-2x + y + 4)

Next, we combine the like terms by adding or subtracting their coefficients. We have three pairs of like terms: x and -2x, -2y and y, and 3 and 4. To combine them, we add their coefficients: 1 + (-2) = -1, -2 + 1 = -1, and 3 + 4 = 7. So, we have:

(x - 2y + 3) + (-2x + y + 4) = -x - y + 7

To practice subtracting linear expressions, you can try some online exercises on IXL or watch some videos on YouTube. You can also use Bitbucket to collaborate with other students and share your work.

In this lesson, we have learned how to subtract linear expressions by distributing the negative sign, combining like terms, and using the distributive property. We have also practiced some examples of how to apply these skills.

Subtracting linear expressions is an important skill that can help us simplify algebraic expressions and solve equations. It can also help us model real-world situations that involve subtraction of quantities with different units.

For example, suppose we want to find the difference between the perimeter of two rectangles. The perimeter of a rectangle is given by the formula P = 2l + 2w, where l is the length and w is the width. If the first rectangle has a length of x + 3 units and a width of x - 2 units, and the second rectangle has a length of 2x - 1 units and a width of x + 1 units, we can subtract their perimeters by subtracting their linear expressions:

Example: Find the difference between the perimeters of two rectangles with dimensions (x + 3, x - 2) and (2x - 1, x + 1).

Solution: To find the difference between the perimeters, we need to subtract their formulas. So, we have:

P_1 - P_2 = (2(x + 3) + 2(x - 2)) - (2(2x - 1) + 2(x + 1))

Next, we need to use the distributive property to multiply each term inside the parenthesis by 2. So, we have:

P_1 - P_2 = (2x + 6 + 2x - 4) - (4x - 2 + 2x + 2)

Next, we need to distribute the negative sign to the second expression. This means we change the sign of each term in the second expression. So, we have:

P_1 - P_2 = (2x + 6 + 2x - 4) + (-4x + 2 - 2x - 2)

Next, we combine the like terms by adding or subtracting their coefficients. We have four pairs of like terms: 2x and -4x, 6 and 2, 2x and -2x, and -4 and -2. To combine them, we add their coefficients: 2 + (-4) = -2, 6 + 2 = 8, 2 + (-2) = 0, and -4 + (-2) = -6. So, we have:

P_1 - P_2 = -2x + 8 + 0x - 6

This simplifies to:

P_1 - P_2 = -2x + 2

This is our final answer. It tells us that the difference between the perimeters of the two rectangles is -2x + 2 units.

In this lesson, we have also learned some key vocabulary and concepts related to linear expressions. Here are some definitions and examples to help you remember them:

Linear expression: An algebraic expression that involves only constants and variables raised to the first power. For example, 3x - 5 and 2y + 7 are linear expressions, but x^2 - 4 and y / 2 are not.

Like terms: Terms that have the same variable and the same exponent. For example, 3x and -2x are like terms, but 3x and 2y are not.

Coefficient: The number in front of the variable. For example, the coefficient of 3x is 3, and the coefficient of -2x is -2.

Distributive property: A property that states that a(b + c) = ab + ac. This means we can multiply a term outside a parenthesis by each term inside the parenthesis and add the results.

Perimeter: The distance around a shape. For a rectangle, the perimeter is given by the formula P = 2l + 2w, where l is the length and w is the width.

You can use these definitions and examples to review what you have learned and to check your understanding of subtracting linear expressions.

In addition to subtracting linear expressions, we can also add, multiply, and divide them. These operations follow similar rules and procedures as subtracting linear expressions. Here are some examples of how to perform these operations:

Example: Add (2x - 3y + 5) + (x + 2y - 7).

Solution: To add two linear expressions, we need to combine the like terms by adding their coefficients. We have three pairs of like terms: 2x and x, -3y and 2y, and 5 and -7. To combine them, we add their coefficients: 2 + 1 = 3, -3 + 2 = -1, and 5 + (-7) = -2. So, we have:

(2x - 3y + 5) + (x + 2y - 7) = 3x - y - 2

Example: Multiply (3x - 4) * 2.

Solution: To multiply a linear expression by a constant, we need to use the distributive property. This means we multiply the constant by each term in the linear expression and add the results. So, we have:

(3x - 4) * 2 = (3x * 2) + (-4 * 2)

Next, we simplify the multiplication by multiplying the coefficients. So, we have:

(3x * 2) + (-4 * 2) = 6x - 8

Example: Divide (6x - 12) / 3.

Solution: To divide a linear expression by a constant, we need to use the distributive property. This means we divide each term in the linear expression by the constant and add the results. So, we have:

(6x - 12) / 3 = (6x / 3) + (-12 / 3)

Next, we simplify the division by dividing the coefficients. So, we have:

(6x / 3) + (-12 / 3) = 2x - 4

In conclusion, we have learned how to subtract linear expressions by following these steps:

Distribute the negative sign to the second expression.

Combine the like terms by adding or subtracting their coefficients.

Use the distributive property if needed to simplify the expressions.

We have also learned some key vocabulary and concepts related to linear expressions, such as terms, like terms, coefficients, and distributive property. We have also practiced some examples of how to add, multiply, and divide linear expressions. We have also seen how to use linear expressions to model real-world situations involving subtraction of quantities with different units.