Verified Answer :

  • Combine the terms containing \(c\) by adding their coefficients: \((0.9 + 0.5)c = 1.4c\).
  • Set the resulting expression equal to the constant: \(1.4c = 7\).
  • Divide both sides by \(1.4\) to isolate \(c\): \(c = \frac{7}{1.4}\).
  • Determine the value of \(c\), which is \(\boxed{5}\).

Explanation

  1. Analysis of the Equation

The problem asks to solve the linear equation \(0.9c + 0.5c = 7\) for the variable \(c\). A linear equation with one variable can be solved by combining like terms and isolating the variable. In this case, both terms on the left-hand side contain the variable \(c\), allowing us to simplify the expression directly.

  1. Combining Like Terms

We start by combining the coefficients of \(c\) on the left side:
\[0.9c + 0.5c = (0.9 + 0.5)c = 1.4c\]
Substituting this back into the original equation gives:
\[1.4c = 7\]

  1. Solving for c

To isolate \(c\), we divide both sides of the equation by \(1.4\):
\[c = \frac{7}{1.4}\]
Performing the division:
\[c = \frac{70}{14} = 5\]
Thus, the value of \(c\) that satisfies the equation is \(5\). We can verify this by plugging \(c = 5\) back into the original equation: \(0.9(5) + 0.5(5) = 4.5 + 2.5 = 7\), which confirms our result is correct.

Examples

Imagine you are mixing two different concentrations of a saline solution. You have a 90% salt solution and a 50% salt solution. If you use a certain amount of the first and an equal amount of the second, and you want to achieve a specific total amount of pure salt (in this case, 7 units), you can set up an equation like this. Solving for the volume of the solutions helps chemists and lab technicians ensure that experiments are prepared with the correct concentrations.

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