Linear equations and inequalities (two variables)

High School Math 1 · Unit 5 · linear-forms-half-planes · Teacher edition

linear-forms-half-planes

Name ______________________________Date ______________

1.
What is the slope of the line $3x + 4y = 12$ ?
Answer: ______________
2.
Write the equation of the line through $(2, -1)$ that is perpendicular to $y = 2x + 4$ . Give it in slope-intercept form, as an equation beginning with $y =$ .
Answer: ______________
3.
A line passes through $(1, 5)$ and $(4, -4)$ . What is its $y$ -intercept?
Answer: ______________
4.
The inequality $y < 2x - 3$ is graphed with a dashed boundary line and one half-plane shaded. Which point lies in the shaded region?
1. (A)
  $(0, 0)$
2. (B)
  $(1, 2)$
3. (C)
  $(2, 1)$
4. (D)
  $(3, 1)$
5.
A line is parallel to $4x - 3y = 9$ and passes through $(6, 1)$ . What is the $y$ -intercept of this line?
Answer: ______________

Answer key — Linear equations and inequalities (two variables)

1.
-3/4linear-forms-half-planes-01
$4y = -3x + 12 \Rightarrow y = -\frac{3}{4}x + 3$ . Slope $= \mathbf{-\dfrac{3}{4}}$ . (Shortcut for $Ax + By = C$ : slope $= -\frac{A}{B}$ .)
2.
$y=-\frac{1}{2}x$ linear-forms-half-planes-02
Perpendicular slope: $-\frac{1}{2}$ . Point-slope: $y + 1 = -\frac{1}{2}(x - 2) = -\frac{1}{2}x + 1$ . Slope-intercept: $\mathbf{y = -\dfrac{1}{2}x}$ — the line passes through the origin. Check: $x = 2$ gives $y = -1$ . ✓
3.
8linear-forms-half-planes-03
$m = \frac{-9}{3} = -3$ . Using $(1, 5)$ : $5 = -3(1) + b \Rightarrow b = 8$ . The line is $y = -3x + 8$ , with $y$ -intercept $\mathbf{8}$ . Check: $x = 4$ gives $-12 + 8 = -4$ . ✓
4.
(D)
$(3, 1)$
linear-forms-half-planes-04
$(3, 1)$ : $1 < 2(3) - 3 = 3$ is true, so $\mathbf{(3, 1)}$ is in the region. ( $(0,0)$ gives $0 < -3$ , false; $(1,2)$ gives $2 < -1$ , false; $(2,1)$ lies on the dashed boundary, which a strict inequality excludes.)
5.
-7linear-forms-half-planes-05
Slope of the given line: $\frac{4}{3}$ . New line: $1 = \frac{4}{3}(6) + b = 8 + b$ , so $b = \mathbf{-7}$ . The line is $y = \frac{4}{3}x - 7$ . (The given line's own intercept, $-3$ , is a different line.)