Answer :
To solve the problem of finding the cost of one square foot of grass sod and one geranium, we need to set up a system of equations based on the information given.
### Step-by-Step Solution:
1. Identify the Variables:
- Let [tex]\( x \)[/tex] represent the cost of one square foot of grass sod.
- Let [tex]\( y \)[/tex] represent the cost of one geranium.
2. Write the Equations:
- Jennifer's Spending:
Jennifer bought [tex]\(9\)[/tex] square feet of grass sod and [tex]\(9\)[/tex] geraniums for a total of [tex]\(\$99\)[/tex].
This can be expressed as the equation:
[tex]\[
9x + 9y = 99
\][/tex]
- Natalie's Spending:
Natalie bought [tex]\(10\)[/tex] square feet of grass sod and [tex]\(3\)[/tex] geraniums for a total of [tex]\(\$82\)[/tex].
This can be expressed as the equation:
[tex]\[
10x + 3y = 82
\][/tex]
3. Solve the System of Equations:
- We have the equations:
[tex]\[
9x + 9y = 99
\][/tex]
[tex]\[
10x + 3y = 82
\][/tex]
4. Graphing Method:
Plotting these equations on a graph, the lines will intersect at a certain point. The x-coordinate of the intersection represents the cost of grass sod, and the y-coordinate represents the cost of a geranium. Through calculation, the intersection of these lines is determined.
5. Solution:
- The point where the two graphs intersect is [tex]\((7, 4)\)[/tex].
6. Conclusion:
- The cost of one square foot of grass sod is [tex]\( \$7 \)[/tex].
- The cost of one geranium is [tex]\( \$4 \)[/tex].
Thus, you can determine the cost of each item by graphing the system of equations and finding where the lines intersect.
### Step-by-Step Solution:
1. Identify the Variables:
- Let [tex]\( x \)[/tex] represent the cost of one square foot of grass sod.
- Let [tex]\( y \)[/tex] represent the cost of one geranium.
2. Write the Equations:
- Jennifer's Spending:
Jennifer bought [tex]\(9\)[/tex] square feet of grass sod and [tex]\(9\)[/tex] geraniums for a total of [tex]\(\$99\)[/tex].
This can be expressed as the equation:
[tex]\[
9x + 9y = 99
\][/tex]
- Natalie's Spending:
Natalie bought [tex]\(10\)[/tex] square feet of grass sod and [tex]\(3\)[/tex] geraniums for a total of [tex]\(\$82\)[/tex].
This can be expressed as the equation:
[tex]\[
10x + 3y = 82
\][/tex]
3. Solve the System of Equations:
- We have the equations:
[tex]\[
9x + 9y = 99
\][/tex]
[tex]\[
10x + 3y = 82
\][/tex]
4. Graphing Method:
Plotting these equations on a graph, the lines will intersect at a certain point. The x-coordinate of the intersection represents the cost of grass sod, and the y-coordinate represents the cost of a geranium. Through calculation, the intersection of these lines is determined.
5. Solution:
- The point where the two graphs intersect is [tex]\((7, 4)\)[/tex].
6. Conclusion:
- The cost of one square foot of grass sod is [tex]\( \$7 \)[/tex].
- The cost of one geranium is [tex]\( \$4 \)[/tex].
Thus, you can determine the cost of each item by graphing the system of equations and finding where the lines intersect.
