Answer:
1. [tex]x-2y=0[/tex]
2. [tex]4x-y=-6[/tex]
3. [tex]5x-2y=-2[/tex]
4. [tex]3x+y=-3[/tex]
Step-by-step explanation:
EQUATIONS OF A LINE:
• Standard form
[tex]Ax+By=C[/tex]
where:
A, B, C = any integer (A is always positive)
• Slope-intercept form: (Use when slope and y-intercept are given)
[tex]y=mx+b[/tex]
where:
m = slope of the line
b = y-intercept
• Point-slope form: (Use when slope and a point are given)
[tex]y-y_1=m(x-x_1)[/tex]
where:
y₁ = y coordinate of the given point
x₁ = x coordinate of the given point
m = slope of the line
• Two-point form: (Use when two points are given)
[tex]y-y_1=\dfrac{y_2-y_1}{x_2-x_1}(x-x_1)[/tex]
where:
y₁ = y coordinate of the given point
x₁ = x coordinate of the given point
y₂ = y coordinate of the given point
x₂ = x coordinate of the given point
• Intercept form: (Use when x and y intercepts are given)
[tex]\dfrac{x}{a}+\dfrac{y}{b}=1[/tex]
where:
a = x-intercept
b = y-intercept
[tex]\\[/tex]
SOLUTION:
[tex]\\[/tex]1.
Given:
Point (2, 1) and slope of 1/2
y₁ = 1
x₁ = 2
m = 1/2
[tex]\\[/tex]
Using point-slope form:
[tex]y-y_1=m(x-x_1)[/tex]
[tex]y-1=\dfrac{1}{2}(x-2)[/tex]
Simplify and convert to standard form
[tex]y-1=\dfrac{1}{2}x-\dfrac{1}{2}(2)[/tex]
[tex]y-1=\dfrac{1}{2}x-1[/tex]
[tex]y=\dfrac{1}{2}x-1+1[/tex]
[tex]y=\dfrac{1}{2}x[/tex]
multiply both sides by 2
[tex]2(y)=2\left(\dfrac{1}{2}x\right)[/tex]
[tex]2y=x[/tex]
transpose 2y to the left side
[tex]0=x-2y[/tex]
Rearrange
[tex]x-2y=0[/tex] (ANSWER)
[tex]\\[/tex]
2.
Given:
slope, m = 4
y-intercept, b = 6
[tex]\\[/tex]
Using the slope-intercept form
[tex]y=mx+b[/tex]
[tex]y=4x+6[/tex]
Transpose y to the left and 6 to the right
[tex]-6=4x-y[/tex]
Rearrange
[tex]4x-y=-6[/tex] (ANSWER)
[tex]\\[/tex]
3.
Given:
Points (0, 1) and (-2, -4)
x₁ = 0
y₁ = 1
x₂ = -2
y₂ = - 4
[tex]\\[/tex]
Using the two-point form of a line:
[tex]y-y_1=\dfrac{y_2-y_1}{x_2-x_1}(x-x_1)[/tex]
[tex]y-1=\dfrac{-4-1}{-2-0}(x-0)[/tex]
simplify and convert to standard form
[tex]y-1=\dfrac{-5}{-2}x[/tex]
[tex]y-1=\dfrac{5}{2}x[/tex]
multiply both sides by 2
[tex]2(y-1)=2\left(\dfrac{5}{2}x\right)[/tex]
[tex]2y-2=5x[/tex]
transpose 2y to the left
[tex]-2=5x-2y[/tex]
rearrange
[tex]5x-2y=-2[/tex] (ANSWER)
[tex]\\[/tex]
4.
Given:
x-intercept, a = -1
y-intercept, b = -3
[tex]\\[/tex]
Using the intercept form of a line:
[tex]\dfrac{x}{a}+\dfrac{y}{b}=1[/tex]
[tex]\dfrac{x}{-1}+\dfrac{y}{-3}=1[/tex]
simplify (addition of fraction with different denominator)
[tex]\dfrac{-3(x)+-1(y)}{-1(-3)}=1[/tex]
[tex]\dfrac{-3x-y}{3}=1[/tex]
multiply both sides by 3
[tex]3\left(\dfrac{-3x-y}{3}\right)=3(1)[/tex]
[tex]-3x-y=3[/tex]
transpose -3x - y to the left and 3 to the right to make the coefficent of x positive
[tex]-3=3x+y[/tex]
rearrange
[tex]3x+y=-3[/tex] (ANSWER)
