The sum of two numbers is 83. If twice the larger number is added to three times the smaller number, the result is 197. Find the numbers.

The two lines are parallel with equal slopes, will never intersect, and have different y-intercepts, showing a constant vertical distance.

To draw a line through each point using the given slope, we will use the slope-point form of a line equation, which is [tex]\( y - y_1 = m(x - x_1) \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( (x_1, y_1) \)[/tex] is the point through which the line passes.

The slope given is [tex]\( \frac{1}{2} \)[/tex]. We have two points, [tex]\( (-2, 1) \)[/tex] and [tex]\( (4, -2) \)[/tex]. Let's calculate the equation of the line for each point.

For the point [tex]\( (-2, 1) \)[/tex]:

[tex]\[ y - 1 = \frac{1}{2}(x - (-2)) \][/tex]

[tex]\[ y - 1 = \frac{1}{2}(x + 2) \][/tex]

[tex]\[ y = \frac{1}{2}x + 2 \cdot \frac{1}{2} + 1 \][/tex]

[tex]\[ y = \frac{1}{2}x + 2 \][/tex]

For the point [tex]\( (4, -2) \)[/tex]:

[tex]\[ y - (-2) = \frac{1}{2}(x - 4) \][/tex]

[tex]\[ y + 2 = \frac{1}{2}x - 4 \cdot \frac{1}{2} \][/tex]

[tex]\[ y = \frac{1}{2}x - 2 - 2 \][/tex]

[tex]\[ y = \frac{1}{2}x - 4 \][/tex]

Now we have the equations for both lines, plot them on the graph to show the relationship between them. Let's proceed with plotting these lines.

(GRAPH IS GIVEN BELOW)

The graph shows two lines:

1. The red line passes through the point [tex]\((-2, 1)\)[/tex] and has a slope of [tex]\(\frac{1}{2}\)[/tex]. Its equation is [tex]\(y = \frac{1}{2}x + 2\)[/tex].

2. The blue line passes through the point [tex]\((4, -2)\)[/tex] and also has a slope of [tex]\(\frac{1}{2}\)[/tex]. Its equation is [tex]\(y = \frac{1}{2}x - 4\)[/tex].

Observing the two lines on the graph, you can notice the following points:

1. Both lines have the same slope of [tex]\(\frac{1}{2}\)[/tex], which means they are parallel to each other.

2. Because they are parallel, the two lines will never intersect each other.

3. The lines have different y-intercepts, which is consistent with parallel lines that are not coincident.

4. The difference in the y-intercepts 2 for the first line and [tex]\(-4\)[/tex] for the second line) corresponds to the vertical distance between the two lines.

To convert 3.27 to a mixed number, separate the whole number (3) from the decimal (0.27), convert the decimal to the fraction 27/100, and combine them to get 3 27/100. The fraction is already in its simplest form.

Converting 3.27 to a Mixed Number in Simplest Form

→ To convert the decimal 3.27 to a mixed number, follow these steps:

Separate the whole number and the decimal part: The whole number is 3, and the decimal part is 0.27.

→ Convert the decimal part to a fraction: 0.27 can be written as 27/100 because there are two decimal places, which means the denominator is 100.

→ Simplify the fraction: To simplify 27/100, find the greatest common divisor (GCD) of 27 and 100, which is 1, meaning the fraction is already in its simplest form.

→ Combine the whole number and the fraction: The final mixed number is 327/100.

Therefore, 3.27 as a mixed number in simplest form is 327/100.

To find the equivalent expression to 8n + 5n, you combine like terms by adding the coefficients of n, resulting in 13n.

The equivalent expression to 8n + 5n is found by combining like terms. Since both terms include the variable n, you can add their coefficients. To find the equivalent expression, you simply add the numbers in front of the n terms:

8n + 5n = (8 + 5)n = 13n.

So, the equivalent expression to 8n + 5n is 13n. This process is based on the distributive property which states that a(b + c) = ab + ac. In this case, we had a common factor of n which we factored out and then added the coefficients 8 and 5.

Final answer:

The equivalent expression is 13n.

Explanation:

The equivalent expression to 8n + 5n is 13n.

To find the equivalent expression, we simply combine the coefficients of n. In this case, 8n represents 8 times n and 5n represents 5 times n. Adding these two expressions together gives us 8n + 5n = 13n.

A double plus fact for 7 is found by using the closest doubles fact to 7 and adjusting one addend slightly, such as 3 + 4 = 7.

A double plus fact for the sum of 7 involves using the concept of doubles in addition. A doubles fact is a simple arithmetic addition where a number is added to itself. For example, 3 + 3 = 6 is a doubles fact.

To create a double plus fact for the sum of 7, we need to consider adding a number to a slightly different number close to a double fact. Here’s a step-by-step approach:

1. Identify the closest doubles fact to 7. In this case, 3 + 3 = 6.

2. Adjust one addend by 1 to reach the sum of 7. So, 3 + 4 = 7.

Therefore, a double plus fact for the sum of 7 is 3 + 4 = 7

To compare the amount of yogurt in a 2 pound container and five 6-ounce containers, convert pounds to ounces. Then compare the measurements. To calculate how many yogurts can be bought for 5 pounds, divide 5 pounds by the price of 1 yogurt.

The Miller's have 3/8 of their income left after setting aside amounts for fixed expenses and savings.

The millers budget 1/2 of their income for fixed expenses and 1/8 of their income for savings. Together, these fractions add up to 1/2 + 1/8 = 5/8 of their income. Calculating what's left of their income involves subtracting the total fraction used (5/8) from the whole (which in terms of fractions, is represented by 1). Hence:

1 - 5/8 = 3/8.

So, the Miller's have 3/8 of their income left after considering fixed expenses and savings.

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