Write a system of equations to describe the situation, solve using an augmented matrix, and fill in the blanks. Khalil decorated every bush with ___ strands of lights and every tre... Write a system of equations to describe the situation, solve using an augmented matrix, and fill in the blanks. Khalil decorated every bush with ___ strands of lights and every tree with ___ strands.
Understand the Problem
The question is asking for the formulation of a system of equations based on the given scenario about Khalil decorating with strands of lights. We'll derive the equations and then solve them using an augmented matrix to find out how many strands of lights were used on each bush and tree.
Answer
Khalil decorated every bush with $5$ strands and every tree with $12$ strands.
Answer for screen readers
Khalil decorated every bush with $5$ strands of lights and every tree with $12$ strands.
Steps to Solve
- Define the Variables Let:
- ( b ): number of strands of lights used on each bush
- ( t ): number of strands of lights used on each tree
- Set Up the Equations Based on the information provided:
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In the morning, Khalil used ( 12 ) strands of lights for ( 1 ) tree. This gives us our first equation: $$ t = 12 $$
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In the afternoon, he used ( 5 ) bushes and ( 1 ) tree for a total of ( 37 ) strands of lights. This gives us our second equation: $$ 5b + t = 37 $$
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Substitute the First Equation into the Second Substituting ( t = 12 ) from the first equation into the second equation: $$ 5b + 12 = 37 $$
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Solve for the Variable ( b ) Now, solve for ( b ): $$ 5b = 37 - 12 $$ $$ 5b = 25 $$ $$ b = 5 $$
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Determine Total Stands of Lights Now we have:
- The number of strands for each bush ( b = 5 )
- The number of strands for each tree ( t = 12 )
Khalil decorated every bush with $5$ strands of lights and every tree with $12$ strands.
More Information
Khalil's decoration totals demonstrate how simple linear equations can be used to solve real-world problems involving total quantities. This example shows the practicality of systems of equations.
Tips
- A common mistake might be confusing the number of bushes and trees or mixing up the values when writing the equations. Always ensure to define your variables clearly.
- Another mistake could be miscalculating the total when substituting the known values.
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