Answered

Find the value of $x$ so that the function has the given value.


  1. $h(x)=-7 x ; h(x)=63$
  2. $t(x)=3 x ; t(x)=24$
  3. $m(x)=4 x+15 ; m(x)=7$
  4. $k(x)=6 x-12 ; k(x)=15$
  5. $q(x)=0.5 x-3 ; q(x)=-4$
  6. $j(x)=-\frac{4}{5} x+7 ; j(x)=-5$

Verified Answer :

  • Set each function expression equal to the target value and isolate the variable \(x\).
  • Solve equation 11: \(x = -9\); equation 12: \(x = 8\); equation 13: \(x = -2\).
  • Solve equation 14: \(x = 4.5\); equation 15: \(x = -2\); equation 16: \(x = 15\).
  • Final results: \(\boxed{x=-9, 8, -2, 4.5, -2, 15}\)

Explanation

  1. Problem Analysis and Strategy

To find the value of \(x\) for each given function, we set the expression for the function equal to its given value. This creates a linear equation in the form \(ax + b = y\). We can solve for \(x\) by isolating the variable through inverse operations: first by subtracting the constant term (if present) from both sides, and then by dividing by the coefficient of \(x\).

  1. Step-by-Step Solving of Linear Equations
  2. For \(h(x) = -7x = 63\), we divide both sides by -7:

\[x = \frac{63}{-7} = -9\]

  1. For \(t(x) = 3x = 24\), we divide both sides by 3:

\[x = \frac{24}{3} = 8\]

  1. For \(m(x) = 4x + 15 = 7\), we first subtract 15 from both sides:

\[4x = 7 - 15 = -8\]
Then divide by 4:
\[x = \frac{-8}{4} = -2\]

  1. For \(k(x) = 6x - 12 = 15\), we first add 12 to both sides:

\[6x = 15 + 12 = 27\]
Then divide by 6:
\[x = \frac{27}{6} = 4.5\]

  1. For \(q(x) = 0.5x - 3 = -4\), we add 3 to both sides:

\[0.5x = -4 + 3 = -1\]
Then divide by 0.5 (which is the same as multiplying by 2):
\[x = \frac{-1}{0.5} = -2\]

  1. For \(j(x) = -\frac{4}{5}x + 7 = -5\), we subtract 7 from both sides:

\[-\frac{4}{5}x = -5 - 7 = -12\]
Then multiply both sides by \(-\frac{5}{4}\):
\[x = -12 \times \left(-\frac{5}{4}\right) = 3 \times 5 = 15\]

  1. Conclusion of Results

The results for the six equations are:

  1. \(x = -9\)
  2. \(x = 8\)
  3. \(x = -2\)
  4. \(x = 4.5\)
  5. \(x = -2\)
  6. \(x = 15\)

Examples

Linear functions are used in many daily situations, such as calculating the cost of a utility service. For example, if a company charges a fixed monthly fee of \(\$15\) and an additional \(\$4\) per unit of energy used, the total cost function is \(C(u) = 4u + 15\). If you received a bill for \(\$7\), you would solve \(4u + 15 = 7\) to determine that you likely have an adjustment or a credit on your account, as the result \(u = -2\) units implies an atypical billing scenario. Understanding how to solve these equations helps in interpreting and managing your financial and resource usage data.

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