How many solutions does this equation have?
[tex]$-3(j+5)=-15-3 j$[/tex]
Answer (2)
Expand the left side of the equation: − 3 ( j + 5 ) = − 3 j − 15 .
Rewrite the equation: − 3 j − 15 = − 15 − 3 j .
Add 3 j to both sides: − 15 = − 15 .
Since the equation is always true, there are infinitely many solutions. in f ini t e l y man y
Explanation
Analyzing the equation We are given the equation − 3 ( j + 5 ) = − 15 − 3 j and asked to determine the number of solutions.
Expanding the left side First, let's expand the left side of the equation by distributing the − 3 across the terms inside the parentheses: − 3 ( j + 5 ) = − 3 × j + ( − 3 ) × 5 = − 3 j − 15
Rewriting the equation Now, we can rewrite the original equation with the expanded left side: − 3 j − 15 = − 15 − 3 j
Adding 3j to both sides Next, we want to isolate the variable j . Let's add 3 j to both sides of the equation: − 3 j − 15 + 3 j = − 15 − 3 j + 3 j This simplifies to: − 15 = − 15
Determining the number of solutions Notice that the variable j has been eliminated from the equation, and we are left with a statement that is always true, − 15 = − 15 . This means that the original equation is true for any value of j . Therefore, the equation has infinitely many solutions.
Examples
Imagine you're trying to balance a budget where some expenses depend on a variable 'j'. If, after simplifying, you find that the equation representing your budget always holds true no matter the value of 'j', it means you have infinite flexibility in adjusting that variable without affecting the overall balance. This is similar to having a recipe where you can adjust one ingredient freely because its effect is always counteracted by another, ensuring the final dish is always perfect.
The equation − 3 ( j + 5 ) = − 15 − 3 j has infinitely many solutions because it simplifies to a true statement, − 15 = − 15 , which holds for any value of j . Therefore, j can take on any real number. The solution is in f ini t e l y man y .
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