Write two equations:

Question

#1 The difference between the two numbers is -10.
#2 The sum of the two numbers is -4.
Let x = _____________
Let y =____________

Answer 1

We can express the two equations based on the statements provided.

1. The difference between the two numbers is -10: \[ x - y = -10 \]

2. The sum of the two numbers is -4: \[ x + y = -4 \]

Now to write it in the requested format:

Let \( x = \) (the first number)
Let \( y = \) (the second number)

So, the complete set becomes:

Let \( x = \) the first number
Let \( y = \) the second number

The equations are:

1. \( x - y = -10 \)
2. \( x + y = -4 \)

Answer 2

Write two equations:
#1 Jack and his sister, Malonie, are 4 years apart in age.
#2 The sum of their ages is 28.

What are their ages?
Let ___ = _____________
Let ___=______________

Answer 3

Let's define the variables for their ages:

Let \( j = \) Jack's age
Let \( m = \) Malonie's age

Now we can write the equations based on the provided statements:

1. Jack and his sister, Malonie, are 4 years apart in age: \[ j - m = 4 \quad \text{(this means Jack is older)} \] Alternatively, it can be expressed as: \[ m - j = 4 \quad \text{(if Malonie is older)} \]

2. The sum of their ages is 28: \[ j + m = 28 \]

To summarize:

Let \( j = \) Jack's age
Let \( m = \) Malonie's age

The equations are:

1. \( j - m = 4 \) (assuming Jack is older)
2. \( j + m = 28 \)

Now, to solve for their ages, we can use the two equations:

From the first equation, we can express \( j \) in terms of \( m \): \[ j = m + 4 \]

Now substitute \( j \) in the second equation: \[ (m + 4) + m = 28 \]

Combine like terms: \[ 2m + 4 = 28 \]

Subtract 4 from both sides: \[ 2m = 24 \]

Divide by 2: \[ m = 12 \]

Now substitute \( m \) back to find \( j \): \[ j = m + 4 = 12 + 4 = 16 \]

Thus, Jack's age is 16 and Malonie's age is 12.

Final answer: Jack's age: 16
Malonie's age: 12

Answer 4

Write two equations:
#1 Two of Julie’s textbooks are a total of $65.
#2 The difference in price between the two books is $9.

What is the cost of each book?
Let ___ = _____________
Let ___=______________

Answer 5

Let's define the variables for the prices of the textbooks:

Let \( x = \) the cost of the first textbook
Let \( y = \) the cost of the second textbook

Now we can write the equations based on the provided statements:

1. Two of Julie’s textbooks are a total of $65: \[ x + y = 65 \]

2. The difference in price between the two books is $9: \[ x - y = 9 \quad \text{(assuming the first book is more expensive)} \]

Now we can summarize:

Let \( x = \) the cost of the first textbook
Let \( y = \) the cost of the second textbook

The equations are:

1. \( x + y = 65 \)
2. \( x - y = 9 \)

Solving the equations

We can use the first equation to express \( y \) in terms of \( x \): \[ y = 65 - x \]

Now, substitute \( y \) in the second equation: \[ x - (65 - x) = 9 \]

Simplify: \[ x - 65 + x = 9 \]

Combine like terms: \[ 2x - 65 = 9 \]

Add 65 to both sides: \[ 2x = 74 \]

Divide by 2: \[ x = 37 \]

Now substitute \( x \) back to find \( y \): \[ y = 65 - x = 65 - 37 = 28 \]

Final Answer:

The cost of the first textbook is $37, and the cost of the second textbook is $28.

So, Let \( x = 37 \) (cost of the first textbook)
Let \( y = 28 \) (cost of the second textbook)

Answer 6

Write a system of equations where the product of two numbers is 10 and the first number is 3 more than the second.
Can you determine the solution? Guess and check.

Answer 7

Let's define the variables:

Let \( x = \) the first number
Let \( y = \) the second number

Now we can write the system of equations based on the given information:

1. The product of two numbers is 10: \[ xy = 10 \]

2. The first number is 3 more than the second: \[ x = y + 3 \]

System of Equations:

1. \( xy = 10 \)
2. \( x = y + 3 \)

Substitute and Solve using Guess and Check

We can substitute the second equation into the first equation:

Substituting \( x \) from the second equation into the first: \[ (y + 3)y = 10 \]

Expanding this gives: \[ y^2 + 3y - 10 = 0 \]

To solve this quadratic equation using guess and check, we can look for two numbers that multiply to \(-10\) and add up to \(3\).

Let's try some values:

1. If \( y = 2 \): \[ x = 2 + 3 = 5 \] Calculate the product: \[ 5 \cdot 2 = 10 \quad (\text{This works!}) \]

2. If \( y = -5 \) (checking for completeness): \[ x = -5 + 3 = -2 \] Calculate the product: \[ -2 \cdot -5 = 10 \quad (\text{This also works!}) \]

Solutions:

1. The numbers can be \( (5, 2) \) or \( (-2, -5) \).

Thus, the solutions to the system of equations are:

• First solution: \( x = 5 \) and \( y = 2 \)
• Second solution: \( x = -2 \) and \( y = -5 \)

Answer 8

Tell whether the ordered pair is a solution of the equation. Just substitute the given x and y to see if the equation “works”. Write “solution” if it works and “not solution” if it doesn’t.

y=4x + 2; (2,10)

2x + y = 5; (7, 5)

Answer 9

To determine if the ordered pairs are solutions for the given equations, we can substitute the values of \( x \) and \( y \) into each equation and see if both sides are equal.

For the first equation:

Equation: \( y = 4x + 2 \)
Ordered Pair: (2, 10)

Substituting \( x = 2 \) and \( y = 10 \) into the equation: \[ 10 = 4(2) + 2 \] \[ 10 = 8 + 2 \] \[ 10 = 10 \quad (\text{True}) \]

Since both sides are equal, the ordered pair (2, 10) is a solution.

For the second equation:

Equation: \( 2x + y = 5 \)
Ordered Pair: (7, 5)

Substituting \( x = 7 \) and \( y = 5 \) into the equation: \[ 2(7) + 5 = 5 \] \[ 14 + 5 = 5 \] \[ 19 = 5 \quad (\text{False}) \]

Since both sides are not equal, the ordered pair (7, 5) is not a solution.

Summary:

1. For \( y = 4x + 2; (2, 10) \) → solution
2. For \( 2x + y = 5; (7, 5) \) → not solution

Answer 10

Set up a table of values:
Y(1) = 1/2x + 5
Y(2) = -2x -1

x
y(1)=
1/2x+5
y(2)=
-2x - 1
(x,y(1))
(x, y(2))
8

4

0

-2

Answer 11

To create a table of values for the functions \( Y(1) = \frac{1}{2}x + 5 \) and \( Y(2) = -2x - 1 \), we will substitute the specified \( x \) values (8, 4, 0, and -2) into both equations and calculate the corresponding \( y \) values.

Let's compute the values:

1. For \( x = 8 \):

   • \( Y(1) = \frac{1}{2}(8) + 5 = 4 + 5 = 9 \)
   • \( Y(2) = -2(8) - 1 = -16 - 1 = -17 \)

2. For \( x = 4 \):

   • \( Y(1) = \frac{1}{2}(4) + 5 = 2 + 5 = 7 \)
   • \( Y(2) = -2(4) - 1 = -8 - 1 = -9 \)

3. For \( x = 0 \):

   • \( Y(1) = \frac{1}{2}(0) + 5 = 0 + 5 = 5 \)
   • \( Y(2) = -2(0) - 1 = 0 - 1 = -1 \)

4. For \( x = -2 \):

   • \( Y(1) = \frac{1}{2}(-2) + 5 = -1 + 5 = 4 \)
   • \( Y(2) = -2(-2) - 1 = 4 - 1 = 3 \)

Summarizing in a table:

\[ \begin{array}{|c|c|c|} \hline x & y(1) = \frac{1}{2}x + 5 & y(2) = -2x - 1 \ \hline 8 & 9 & -17 \ 4 & 7 & -9 \ 0 & 5 & -1 \ -2 & 4 & 3 \ \hline \end{array} \]

Final Table of Values:

\[ \begin{array}{|c|c|c|} \hline x & y(1) & y(2) \ \hline 8 & 9 & -17 \ 4 & 7 & -9 \ 0 & 5 & -1 \ -2 & 4 & 3 \ \hline \end{array} \]

This table summarizes the calculated \( y \) values for each corresponding \( x \).