Answer :
To solve for the two numbers given the conditions that their sum is 7 and their difference is 21, we can set up a system of linear equations and solve it step-by-step.
Let's denote the two numbers as [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
The two given conditions translate into the following system of equations:
1. [tex]\( x + y = 7 \)[/tex]
2. [tex]\( x - y = 21 \)[/tex]
To find the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex], we can use the method of addition to eliminate one of the variables.
### Step 1: Add the two equations
Adding the two given equations will help us eliminate [tex]\( y \)[/tex]:
[tex]\[ (x + y) + (x - y) = 7 + 21 \][/tex]
Simplifying this, we get:
[tex]\[ x + y + x - y = 28 \][/tex]
[tex]\[ 2x = 28 \][/tex]
### Step 2: Solve for [tex]\( x \)[/tex]
Divide both sides of the equation by 2 to isolate [tex]\( x \)[/tex]:
[tex]\[ x = 14 \][/tex]
### Step 3: Substitute [tex]\( x \)[/tex] back into one of the original equations
To solve for [tex]\( y \)[/tex], we substitute [tex]\( x \)[/tex] back into the first equation [tex]\( x + y = 7 \)[/tex]:
[tex]\[ 14 + y = 7 \][/tex]
### Step 4: Solve for [tex]\( y \)[/tex]
Isolate [tex]\( y \)[/tex] by subtracting 14 from both sides:
[tex]\[ y = 7 - 14 \][/tex]
[tex]\[ y = -7 \][/tex]
### Conclusion
The two numbers are [tex]\( x = 14 \)[/tex] and [tex]\( y = -7 \)[/tex].
Thus, the numbers are 14 and -7.
Let's denote the two numbers as [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
The two given conditions translate into the following system of equations:
1. [tex]\( x + y = 7 \)[/tex]
2. [tex]\( x - y = 21 \)[/tex]
To find the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex], we can use the method of addition to eliminate one of the variables.
### Step 1: Add the two equations
Adding the two given equations will help us eliminate [tex]\( y \)[/tex]:
[tex]\[ (x + y) + (x - y) = 7 + 21 \][/tex]
Simplifying this, we get:
[tex]\[ x + y + x - y = 28 \][/tex]
[tex]\[ 2x = 28 \][/tex]
### Step 2: Solve for [tex]\( x \)[/tex]
Divide both sides of the equation by 2 to isolate [tex]\( x \)[/tex]:
[tex]\[ x = 14 \][/tex]
### Step 3: Substitute [tex]\( x \)[/tex] back into one of the original equations
To solve for [tex]\( y \)[/tex], we substitute [tex]\( x \)[/tex] back into the first equation [tex]\( x + y = 7 \)[/tex]:
[tex]\[ 14 + y = 7 \][/tex]
### Step 4: Solve for [tex]\( y \)[/tex]
Isolate [tex]\( y \)[/tex] by subtracting 14 from both sides:
[tex]\[ y = 7 - 14 \][/tex]
[tex]\[ y = -7 \][/tex]
### Conclusion
The two numbers are [tex]\( x = 14 \)[/tex] and [tex]\( y = -7 \)[/tex].
Thus, the numbers are 14 and -7.
