A mathematics problem uses Slope–intercept line. Given slope (m) = 0.9000; x-value (x) = 5.5000; intercept (b) = -9.0000, 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

   $$List the givens: slope (m) = 0.9000, x-value (x) = 5.5000, intercept (b) = -9.0000$$

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

5. Step 5 — Evaluate:

   $$y = -4.0500$$

6. Step 6 — Check: returning y = -4.0500 to

   $$y = m x + b$$

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

Why the other options are there

• -8.1000 — kept a factor of two that cancels in the correct rearrangement.
• -2.0250 — dropped that same factor in the other direction.
• -4.4550 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Mathematics → Straight Line

A designer checks the slope of a straight line on a site grading plan. Given x1 (x1) = 5.0000; x2 (x2) = 9.0000; y1 (y1) = 6.5000; y2 (y2) = 26.0000, determine the slope (m).

Start with the thinking

• The governing relation printed in this handbook section is Straight Line — two-point slope.
• 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.
• The FE Handbook Straight Line relation gives the slope through two points on a line.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$m = \dfrac{y_2 - y_1}{x_2 - x_1}$$

2. Step 2 — Rearrange symbolically for m:

   $$m = \dfrac{y_2 - y_1}{x_2 - x_1}$$

3. Step 3

   $$List the givens: x_{1} (x_{1}) = 5.0000, x_{2} (x_{2}) = 9.0000, y_{1} (y_{1}) = 6.5000, y_{2} (y_{2}) = 26.0000$$

4. Step 4 — Substitute the given values:

   $$m = \dfrac{y_2 - y_1}{x_2 - x_1}$$

5. Step 5 — Evaluate:

   $$m = 4.8750$$

6. Step 6 — Check: returning m = 4.8750 to

   $$m = \dfrac{y_2 - y_1}{x_2 - x_1}$$

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

Why the other options are there

• 9.7500 — kept a factor of two that cancels in the correct rearrangement.
• 2.4375 — dropped that same factor in the other direction.
• 5.3625 — rounded an intermediate value before the final step.

Reference: FE Handbook — Straight Line

A mathematics problem uses Slope–intercept line. Given x-value (x) = 8.0000; intercept (b) = 9.5000; y-value (y) = 54.4000, 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

   $$List the givens: x-value (x) = 8.0000, intercept (b) = 9.5000, y-value (y) = 54.4000$$

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

5. Step 5 — Evaluate:

   $$m = 5.6125$$

6. Step 6 — Check: returning m = 5.6125 to

   $$y = m x + b$$

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

Why the other options are there

• 11.2250 — kept a factor of two that cancels in the correct rearrangement.
• 2.8063 — dropped that same factor in the other direction.
• 6.1738 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Mathematics → Straight Line

A drafter verifies a straight line segment on a coordinate layout. Given x1 (x1) = 3.5000; x2 (x2) = 13.5000; y1 (y1) = 5.5000; slope (m) = 3.0500, determine the y2 (y2).

Start with the thinking

• The governing relation printed in this handbook section is Straight Line — two-point slope.
• Everything except y2 is given, so isolate y2 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 FE Handbook Straight Line relation gives the slope through two points on a line.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$m = \dfrac{y_2 - y_1}{x_2 - x_1}$$

2. Step 2 — Rearrange symbolically for y2:

   $$y_{2} = m (x_2 - x_1) + y_1$$

3. Step 3

   $$List the givens: x_{1} (x_{1}) = 3.5000, x_{2} (x_{2}) = 13.5000, y_{1} (y_{1}) = 5.5000, slope (m) = 3.0500$$

4. Step 4 — Substitute the given values:

   $$y_{2} = 3.0500 (x_2 - x_1) + y_1$$

5. Step 5 — Evaluate:

   $$y_{2} = 36.0000$$

6. Step 6 — Check: returning y2 = 36.0000 to

   $$m = \dfrac{y_2 - y_1}{x_2 - x_1}$$

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

Why the other options are there

• 72.0000 — kept a factor of two that cancels in the correct rearrangement.
• 18.0000 — dropped that same factor in the other direction.
• 39.6000 — rounded an intermediate value before the final step.

Reference: FE Handbook — Straight Line

A mathematics problem uses Slope–intercept line. Given slope (m) = 2.0000; x-value (x) = 7.0000; y-value (y) = 38.3000, 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

   $$List the givens: slope (m) = 2.0000, x-value (x) = 7.0000, y-value (y) = 38.3000$$

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

5. Step 5 — Evaluate:

   $$b = 24.3000$$

6. Step 6 — Check: returning b = 24.3000 to

   $$y = m x + b$$

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

Why the other options are there

• 48.6000 — kept a factor of two that cancels in the correct rearrangement.
• 12.1500 — dropped that same factor in the other direction.
• 26.7300 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Mathematics → Straight Line

A surveyor plots a straight line between two staked points. Given x1 (x1) = 4.5000; y1 (y1) = 7.0000; y2 (y2) = 25.5000; slope (m) = 2.6500, determine the x2 (x2).

Start with the thinking

• The governing relation printed in this handbook section is Straight Line — two-point slope.
• Everything except x2 is given, so isolate x2 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 FE Handbook Straight Line relation gives the slope through two points on a line.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$m = \dfrac{y_2 - y_1}{x_2 - x_1}$$

2. Step 2 — Rearrange symbolically for x2:

   $$x_{2} = \dfrac{y_2 - y_1}{m} + x_1$$

3. Step 3

   $$List the givens: x_{1} (x_{1}) = 4.5000, y_{1} (y_{1}) = 7.0000, y_{2} (y_{2}) = 25.5000, slope (m) = 2.6500$$

4. Step 4 — Substitute the given values:

   $$x_{2} = \dfrac{y_2 - y_1}{2.6500} + x_1$$

5. Step 5 — Evaluate:

   $$x_{2} = 11.4811$$

6. Step 6 — Check: returning x2 = 11.4811 to

   $$m = \dfrac{y_2 - y_1}{x_2 - x_1}$$

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

Why the other options are there

• 22.9623 — kept a factor of two that cancels in the correct rearrangement.
• 5.7406 — dropped that same factor in the other direction.
• 12.6292 — rounded an intermediate value before the final step.

Reference: FE Handbook — Straight Line

A mathematics problem uses Slope–intercept line. Given slope (m) = 4.9000; intercept (b) = -4.0000; y-value (y) = 42.8000, 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

   $$List the givens: slope (m) = 4.9000, intercept (b) = -4.0000, y-value (y) = 42.8000$$

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

5. Step 5 — Evaluate:

   $$x = 9.5510$$

6. Step 6 — Check: returning x = 9.5510 to

   $$y = m x + b$$

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

Why the other options are there

• 19.1020 — kept a factor of two that cancels in the correct rearrangement.
• 4.7755 — dropped that same factor in the other direction.
• 10.5061 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Mathematics → Straight Line

A designer checks the slope of a straight line on a site grading plan. Given x1 (x1) = 1.5000; x2 (x2) = 11.0000; y1 (y1) = -2.5000; y2 (y2) = 13.0000, determine the slope (m).

Start with the thinking

• The governing relation printed in this handbook section is Straight Line — two-point slope.
• 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.
• The FE Handbook Straight Line relation gives the slope through two points on a line.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$m = \dfrac{y_2 - y_1}{x_2 - x_1}$$

2. Step 2 — Rearrange symbolically for m:

   $$m = \dfrac{y_2 - y_1}{x_2 - x_1}$$

3. Step 3

   $$List the givens: x_{1} (x_{1}) = 1.5000, x_{2} (x_{2}) = 11.0000, y_{1} (y_{1}) = -2.5000, y_{2} (y_{2}) = 13.0000$$

4. Step 4 — Substitute the given values:

   $$m = \dfrac{y_2 - y_1}{x_2 - x_1}$$

5. Step 5 — Evaluate:

   $$m = 1.6316$$

6. Step 6 — Check: returning m = 1.6316 to

   $$m = \dfrac{y_2 - y_1}{x_2 - x_1}$$

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

Why the other options are there

• 3.2632 — kept a factor of two that cancels in the correct rearrangement.
• 0.8158 — dropped that same factor in the other direction.
• 1.7947 — rounded an intermediate value before the final step.

Reference: FE Handbook — Straight Line

A mathematics problem uses Slope–intercept line. Given slope (m) = 1.3000; x-value (x) = 9.5000; intercept (b) = -1.5000, 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

   $$List the givens: slope (m) = 1.3000, x-value (x) = 9.5000, intercept (b) = -1.5000$$

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

5. Step 5 — Evaluate:

   $$y = 10.8500$$

6. Step 6 — Check: returning y = 10.8500 to

   $$y = m x + b$$

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

Why the other options are there

• 21.7000 — kept a factor of two that cancels in the correct rearrangement.
• 5.4250 — dropped that same factor in the other direction.
• 11.9350 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Mathematics → Straight Line

A drafter verifies a straight line segment on a coordinate layout. Given x1 (x1) = 4.0000; x2 (x2) = 10.5000; y1 (y1) = 9.5000; slope (m) = 1.4000, determine the y2 (y2).

Start with the thinking

• The governing relation printed in this handbook section is Straight Line — two-point slope.
• Everything except y2 is given, so isolate y2 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 FE Handbook Straight Line relation gives the slope through two points on a line.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$m = \dfrac{y_2 - y_1}{x_2 - x_1}$$

2. Step 2 — Rearrange symbolically for y2:

   $$y_{2} = m (x_2 - x_1) + y_1$$

3. Step 3

   $$List the givens: x_{1} (x_{1}) = 4.0000, x_{2} (x_{2}) = 10.5000, y_{1} (y_{1}) = 9.5000, slope (m) = 1.4000$$

4. Step 4 — Substitute the given values:

   $$y_{2} = 1.4000 (x_2 - x_1) + y_1$$

5. Step 5 — Evaluate:

   $$y_{2} = 18.6000$$

6. Step 6 — Check: returning y2 = 18.6000 to

   $$m = \dfrac{y_2 - y_1}{x_2 - x_1}$$

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

Why the other options are there

• 37.2000 — kept a factor of two that cancels in the correct rearrangement.
• 9.3000 — dropped that same factor in the other direction.
• 20.4600 — rounded an intermediate value before the final step.

Reference: FE Handbook — Straight Line