A mathematics problem uses Slope–intercept line. Given slope (m) = 2.6000; x-value (x) = 1.5000; intercept (b) = 8.0000, determine the y-value (y).

Start with the thinking

• The governing relation printed in this handbook section is Slope–intercept line.
• Everything except y is given, so isolate y symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Mathematics items reward recognising the unknown before touching a calculator.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$y = m x + b$$

2. Step 2 — Rearrange the relation so that y stands alone on the left-hand side.

3. Step 3

   $$List the givens: slope (m) = 2.6000, x-value (x) = 1.5000, intercept (b) = 8.0000$$

4. Step 4 — Substitute the given values into the rearranged relation.

5. Step 5 — Evaluate:

   $$y = 11.9000$$

6. Step 6 — Check: returning y = 11.9000 to

   $$y = m x + b$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 23.8000 — kept a factor of two that cancels in the correct rearrangement.
• 5.9500 — dropped that same factor in the other direction.
• 13.0900 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Mathematics → First-Order Linear Difference Equation

radioactive tracer decay in a groundwater test Given initial value (y_0) = 10.9000 mg/L; rate constant (k) = 0.5000 1/h; elapsed time (t) = 1.4000 h, determine the value at time t (y) in mg/L.

Start with the thinking

• The governing relation printed in this handbook section is First-order linear ODE solution.
• Everything except y is given, so isolate y symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• The ODE y' + k y = 0 has the exponential solution shown.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$y(t) = y_0 e^{-k t}$$

2. Step 2 — Rearrange symbolically for y:

   $$y = y = y_0 e^{-k t}$$

3. Step 3 — List the givens: initial value (y_0) = 10.9000 mg/L, rate constant (k) = 0.5000 1/h, elapsed time (t) = 1.4000 h.

4. Step 4 — Substitute the given values:

   $$y = y = y_0 e^{-0.5000 1.4000}$$

5. Step 5 — Evaluate:

   $$y = 5.4128\ \text{mg/L}$$

6. Step 6 — Check: returning y = 5.4128 mg/L to

   $$y(t) = y_0 e^{-k t}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 10.8256 — kept a factor of two that cancels in the correct rearrangement.
• 2.7064 — dropped that same factor in the other direction.
• 5.9541 — rounded an intermediate value before the final step.

Reference: FE Handbook — Differential Equations (First Order)

A mathematics problem uses Slope–intercept line. Given x-value (x) = 8.0000; intercept (b) = -9.0000; y-value (y) = -16.7000, determine the slope (m).

Start with the thinking

• The governing relation printed in this handbook section is Slope–intercept line.
• Everything except m is given, so isolate m symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Mathematics items reward recognising the unknown before touching a calculator.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$y = m x + b$$

2. Step 2 — Rearrange the relation so that m stands alone on the left-hand side.

3. Step 3

   $$List the givens: x-value (x) = 8.0000, intercept (b) = -9.0000, y-value (y) = -16.7000$$

4. Step 4 — Substitute the given values into the rearranged relation.

5. Step 5 — Evaluate:

   $$m = -0.9625$$

6. Step 6 — Check: returning m = -0.9625 to

   $$y = m x + b$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• -1.9250 — kept a factor of two that cancels in the correct rearrangement.
• -0.4812 — dropped that same factor in the other direction.
• -1.0588 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Mathematics → First-Order Linear Difference Equation

temperature decay of a mass concrete pour Given value at time t (y) = 3.1000 mg/L; rate constant (k) = 0.5000 1/h; elapsed time (t) = 5.1000 h, determine the initial value (y_0) in mg/L.

Start with the thinking

• The governing relation printed in this handbook section is First-order linear ODE solution.
• Everything except y_0 is given, so isolate y_0 symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• The ODE y' + k y = 0 has the exponential solution shown.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$y(t) = y_0 e^{-k t}$$

2. Step 2 — Rearrange symbolically for y_0:

   $$y_{0} = y_0 = y e^{k t}$$

3. Step 3 — List the givens: value at time t (y) = 3.1000 mg/L, rate constant (k) = 0.5000 1/h, elapsed time (t) = 5.1000 h.

4. Step 4 — Substitute the given values:

   $$y_{0} = y_0 = 3.1000 e^{0.5000 5.1000}$$

5. Step 5 — Evaluate:

   $$y_{0} = 39.7020\ \text{mg/L}$$

6. Step 6 — Check: returning y_0 = 39.7020 mg/L to

   $$y(t) = y_0 e^{-k t}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 79.4040 — kept a factor of two that cancels in the correct rearrangement.
• 19.8510 — dropped that same factor in the other direction.
• 43.6722 — rounded an intermediate value before the final step.

Reference: FE Handbook — Differential Equations (First Order)

A mathematics problem uses Slope–intercept line. Given slope (m) = 2.5000; x-value (x) = 9.0000; y-value (y) = 28.8000, determine the intercept (b).

Start with the thinking

• The governing relation printed in this handbook section is Slope–intercept line.
• Everything except b is given, so isolate b symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Mathematics items reward recognising the unknown before touching a calculator.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$y = m x + b$$

2. Step 2 — Rearrange the relation so that b stands alone on the left-hand side.

3. Step 3

   $$List the givens: slope (m) = 2.5000, x-value (x) = 9.0000, y-value (y) = 28.8000$$

4. Step 4 — Substitute the given values into the rearranged relation.

5. Step 5 — Evaluate:

   $$b = 6.3000$$

6. Step 6 — Check: returning b = 6.3000 to

   $$y = m x + b$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 12.6000 — kept a factor of two that cancels in the correct rearrangement.
• 3.1500 — dropped that same factor in the other direction.
• 6.9300 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Mathematics → First-Order Linear Difference Equation

chlorine residual decay in a storage tank Given value at time t (y) = 3.0000 mg/L; initial value (y_0) = 10.1000 mg/L; elapsed time (t) = 9.5000 h, determine the rate constant (k) in 1/h.

Start with the thinking

• The governing relation printed in this handbook section is First-order linear ODE solution.
• Everything except k is given, so isolate k symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• The ODE y' + k y = 0 has the exponential solution shown.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$y(t) = y_0 e^{-k t}$$

2. Step 2 — Rearrange symbolically for k:

   $$k = k = \dfrac{\ln(y_0 / y)}{t}$$

3. Step 3 — List the givens: value at time t (y) = 3.0000 mg/L, initial value (y_0) = 10.1000 mg/L, elapsed time (t) = 9.5000 h.

4. Step 4 — Substitute the given values:

   $$k = k = \dfrac{\ln(y_0 / 3.0000)}{9.5000}$$

5. Step 5 — Evaluate:

   $$k = 0.1278\ \text{1/h}$$

6. Step 6 — Check: returning k = 0.1278 1/h to

   $$y(t) = y_0 e^{-k t}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 0.2556 — kept a factor of two that cancels in the correct rearrangement.
• 0.0639 — dropped that same factor in the other direction.
• 0.1406 — rounded an intermediate value before the final step.

Reference: FE Handbook — Differential Equations (First Order)

A mathematics problem uses Slope–intercept line. Given slope (m) = 2.6000; intercept (b) = -6.0000; y-value (y) = 19.0000, determine the x-value (x).

Start with the thinking

• The governing relation printed in this handbook section is Slope–intercept line.
• Everything except x is given, so isolate x symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Mathematics items reward recognising the unknown before touching a calculator.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$y = m x + b$$

2. Step 2 — Rearrange the relation so that x stands alone on the left-hand side.

3. Step 3

   $$List the givens: slope (m) = 2.6000, intercept (b) = -6.0000, y-value (y) = 19.0000$$

4. Step 4 — Substitute the given values into the rearranged relation.

5. Step 5 — Evaluate:

   $$x = 9.6154$$

6. Step 6 — Check: returning x = 9.6154 to

   $$y = m x + b$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 19.2308 — kept a factor of two that cancels in the correct rearrangement.
• 4.8077 — dropped that same factor in the other direction.
• 10.5769 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Mathematics → First-Order Linear Difference Equation

radioactive tracer decay in a groundwater test Given value at time t (y) = 2.2000 mg/L; initial value (y_0) = 4.1000 mg/L; rate constant (k) = 0.6000 1/h, determine the elapsed time (t) in h.

Start with the thinking

• The governing relation printed in this handbook section is First-order linear ODE solution.
• Everything except t is given, so isolate t symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• The ODE y' + k y = 0 has the exponential solution shown.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$y(t) = y_0 e^{-k t}$$

2. Step 2 — Rearrange symbolically for t:

   $$t = t = \dfrac{\ln(y_0 / y)}{k}$$

3. Step 3 — List the givens: value at time t (y) = 2.2000 mg/L, initial value (y_0) = 4.1000 mg/L, rate constant (k) = 0.6000 1/h.

4. Step 4 — Substitute the given values:

   $$t = t = \dfrac{\ln(y_0 / 2.2000)}{0.6000}$$

5. Step 5 — Evaluate:

   $$t = 1.0375\ \text{h}$$

6. Step 6 — Check: returning t = 1.0375 h to

   $$y(t) = y_0 e^{-k t}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 2.0751 — kept a factor of two that cancels in the correct rearrangement.
• 0.5188 — dropped that same factor in the other direction.
• 1.1413 — rounded an intermediate value before the final step.

Reference: FE Handbook — Differential Equations (First Order)

A mathematics problem uses Slope–intercept line. Given slope (m) = 1.4000; x-value (x) = 4.0000; intercept (b) = -6.5000, determine the y-value (y).

Start with the thinking

• The governing relation printed in this handbook section is Slope–intercept line.
• Everything except y is given, so isolate y symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Mathematics items reward recognising the unknown before touching a calculator.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$y = m x + b$$

2. Step 2 — Rearrange the relation so that y stands alone on the left-hand side.

3. Step 3

   $$List the givens: slope (m) = 1.4000, x-value (x) = 4.0000, intercept (b) = -6.5000$$

4. Step 4 — Substitute the given values into the rearranged relation.

5. Step 5 — Evaluate:

   $$y = -0.9000$$

6. Step 6 — Check: returning y = -0.9000 to

   $$y = m x + b$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• -1.8000 — kept a factor of two that cancels in the correct rearrangement.
• -0.4500 — dropped that same factor in the other direction.
• -0.9900 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Mathematics → First-Order Linear Difference Equation

temperature decay of a mass concrete pour Given initial value (y_0) = 11.3000 mg/L; rate constant (k) = 0.6000 1/h; elapsed time (t) = 9.4000 h, determine the value at time t (y) in mg/L.

Start with the thinking

• The governing relation printed in this handbook section is First-order linear ODE solution.
• Everything except y is given, so isolate y symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• The ODE y' + k y = 0 has the exponential solution shown.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$y(t) = y_0 e^{-k t}$$

2. Step 2 — Rearrange symbolically for y:

   $$y = y = y_0 e^{-k t}$$

3. Step 3 — List the givens: initial value (y_0) = 11.3000 mg/L, rate constant (k) = 0.6000 1/h, elapsed time (t) = 9.4000 h.

4. Step 4 — Substitute the given values:

   $$y = y = y_0 e^{-0.6000 9.4000}$$

5. Step 5 — Evaluate:

   $$y = 0.0401\ \text{mg/L}$$

6. Step 6 — Check: returning y = 0.0401 mg/L to

   $$y(t) = y_0 e^{-k t}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 0.0803 — kept a factor of two that cancels in the correct rearrangement.
• 0.0201 — dropped that same factor in the other direction.
• 0.0442 — rounded an intermediate value before the final step.

Reference: FE Handbook — Differential Equations (First Order)