Richie spent $\$ 120$ on $12$ rose bushes and $2$ gardenias, while Jim spent $3150$ on $10$ rose bushes and $10$ gardenias. What is the cost of $1$ rose bush and $1$ gardenia?

Full solution

Q. Richie spent

\$ 120

12

rose bushes and

2

gardenias, while Jim spent

3150

10

rose bushes and

10

gardenias. What is the cost of

1

rose bush and

1

gardenia?

Equation for Richie's purchase: Let's denote the cost of one rose bush as $R$ dollars and the cost of one gardenia as $G$ dollars. Richie's purchase can be represented by the equation: $\newline$ $12R + 2G = 120$ dollars.
Equation for Jim's purchase: Similarly, Jim's purchase can be represented by the equation: $10R + 10G = 3150$ dollars.
Elimination method: To solve for $R$ and $G$ , we can use the method of substitution or elimination. Let's use elimination. First, we'll multiply Richie's equation by $5$ to align the $G$ terms with Jim's equation: $\newline$ $(12R + 2G) \times 5 = 120 \times 5$ $\newline$ $60R + 10G = 600$
Subtract equations: Now we have two equations: $\newline$ $60R + 10G = 600$ (Richie's equation multiplied by $5$ ) $\newline$ $10R + 10G = 3150$ (Jim's equation) $\newline$ We can subtract Jim's equation from the modified Richie's equation to eliminate $G$ : $\newline$ $(60R + 10G) - (10R + 10G) = 600 - 3150$ $\newline$ $50R = -2550$
Solve for R: Dividing both sides of the equation by $50$ to solve for $R$ : $\frac{50R}{50} = \frac{-2550}{50}$ $R = -51$ We have a negative value for the cost of a rose bush, which doesn't make sense in this context. There must be a mistake in the previous steps.