Answer :
To solve the real-world system using graphing, let's first write the equations based on the given scenario:
1. Equation for Jennifer:
- Jennifer spent [tex]$99 on \(9 \, \text{ft}^2\) of grass sod and 9 geraniums.
- Let \(x\) be the cost of 1 \( \text{ft}^2\) of grass sod and \(y\) be the cost of 1 geranium.
- This can be written as:
\[ 9x + 9y = 99 \]
- Simplify the equation by dividing everything by 9:
\[ x + y = 11 \]
2. Equation for Natalie:
- Natalie spent $[/tex]82 on [tex]\(10 \, \text{ft}^2\)[/tex] of grass sod and 3 geraniums.
- Write this equation as:
[tex]\[ 10x + 3y = 82 \][/tex]
Now, graph these two linear equations:
- The first equation, [tex]\(x + y = 11\)[/tex], can be rearranged to find [tex]\(y\)[/tex]:
[tex]\[ y = 11 - x \][/tex]
This is a straight line with a y-intercept of 11 and a slope of -1.
- The second equation, [tex]\(10x + 3y = 82\)[/tex], can similarly be rearranged:
[tex]\[ 3y = 82 - 10x \][/tex]
[tex]\[ y = \frac{82 - 10x}{3} \][/tex]
This line has a y-intercept when [tex]\(x = 0\)[/tex] and a slope of [tex]\(-\frac{10}{3}\)[/tex].
To find the point of intersection, graph both lines on the same coordinate axes. The point where the two lines intersect will give us the values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex].
After graphing:
- The point of intersection is [tex]\((7, 4)\)[/tex].
This tells us:
- Cost of 1 [tex]\( \text{ft}^2\)[/tex] of Grass Sod ([tex]\(x\)[/tex]): \[tex]$7
- Cost of 1 Geranium (\(y\)): \$[/tex]4
This intersection point represents the solution to the system of equations, providing the cost of the individual items Jennifer and Natalie purchased.
1. Equation for Jennifer:
- Jennifer spent [tex]$99 on \(9 \, \text{ft}^2\) of grass sod and 9 geraniums.
- Let \(x\) be the cost of 1 \( \text{ft}^2\) of grass sod and \(y\) be the cost of 1 geranium.
- This can be written as:
\[ 9x + 9y = 99 \]
- Simplify the equation by dividing everything by 9:
\[ x + y = 11 \]
2. Equation for Natalie:
- Natalie spent $[/tex]82 on [tex]\(10 \, \text{ft}^2\)[/tex] of grass sod and 3 geraniums.
- Write this equation as:
[tex]\[ 10x + 3y = 82 \][/tex]
Now, graph these two linear equations:
- The first equation, [tex]\(x + y = 11\)[/tex], can be rearranged to find [tex]\(y\)[/tex]:
[tex]\[ y = 11 - x \][/tex]
This is a straight line with a y-intercept of 11 and a slope of -1.
- The second equation, [tex]\(10x + 3y = 82\)[/tex], can similarly be rearranged:
[tex]\[ 3y = 82 - 10x \][/tex]
[tex]\[ y = \frac{82 - 10x}{3} \][/tex]
This line has a y-intercept when [tex]\(x = 0\)[/tex] and a slope of [tex]\(-\frac{10}{3}\)[/tex].
To find the point of intersection, graph both lines on the same coordinate axes. The point where the two lines intersect will give us the values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex].
After graphing:
- The point of intersection is [tex]\((7, 4)\)[/tex].
This tells us:
- Cost of 1 [tex]\( \text{ft}^2\)[/tex] of Grass Sod ([tex]\(x\)[/tex]): \[tex]$7
- Cost of 1 Geranium (\(y\)): \$[/tex]4
This intersection point represents the solution to the system of equations, providing the cost of the individual items Jennifer and Natalie purchased.
