Writing Linear Equations

Writing Linear Equations exercise

• Determine the linear equation in slope-intercept form.

  Hints

  The slope-intercept form is: $y=mx+b$.

  • $m$ is the slope
  • $b$ is the y-intercept

  To calculate the y-intercept, you have to plug in one of the given points.

  When calculating the slope, make sure to be consistent when ordering the points.

  Solution

  A linear equation written in slope-intercept form is: $y=mx+b$.

  $m$ represents the slope, and $b$ represents the y-intercept.

  In order to identify the linear equation, we start with the slope, which is given as the change in y divided by the change in x:

  • $m=\frac{y_2-y_1}{x_2-x_1}$

  Let's plug in the given coordinates of the two points:

  • $m=\frac{8-3}{12-2}=\frac{5}{10}=\frac{5\div 5}{10\div 5}=\frac12$

  The equation is:

  • $y=\frac12x+b$

  The y-intercept $b$ is still unknown. To figure out the y-intercept, we plug the coordinates of either $P$ or $W$ into the equation. Here we use $P$. You can also use $W$, you'll get the same y-intercept value.

  $\begin{align*} 3&=\frac12\times2+b\\ &=1+b \end{align*}$

  Subtracting $1$ leads to $b=2$.

  As a result we get the linear equation in slope-intercept form:

  • $y=\frac12x+2$

  You can check this equation by substituting the coordinates of the other point into the equation:

  • $8=\frac12\times 12+2=6+2$ $\surd$

• Explain how to set up an equation for a parallel line.

  Hints

  Parallel lines always have the same slope.

  A linear equation in slope-intercept form is: $y=mx+b$.

  • $m$ is the slope
  • $b$ is the y-intercept

  You can calculate the y-intercept.

  To figure out the y-intercept, plug the coordinates of the given point into the linear equation. This will result in a value for $b$.

  Solution

  Remember, the slope-intercept form is: $y=mx+b$.

  $m$ is the slope, and $b$ is the y-intercept. Two parallel lines have the same slope.

  • $m=\frac12$

  We still have to identify the y-intercept. To do so, we plug in the coordinates of $P_2(2,4)$ into the equation:

  • $y=\frac12x+b$

  We can calculate:

  • $4=\frac12\times 2+b=1+b$

  Subtracting $1$ results in:

  • $b=3$

  Finally, we identify the linear equation in slope-intercept form:

  • $y=\frac12x+3$

  The two equations are parallel, but not identical, because $3\neq2$. That means the y-intercepts are different.

• Find the equation of the line that is perpendicular to $y=\frac12 x+2$ and passes through $B(7,5)$.

  Hints

  In order to write the linear equation in slope-intercept form, you have to determine the slope and then calculate the y-intercept.

  $y=m_1x+b_1$ and $y=m_2x+b_2$ are two linear equations.

  The lines are perpendicular if $m_1\times m_2=-1$.

  In order to find the y-intercept, you have to plug the coordinates of a given point into the equation $y=mx+b$.

  Solution

  A linear equation in slope-intercept form is:$y=mx+b$.

  The equation of the purple line has to be perpendicular to $y=\frac12x+2$.

  So, $\frac12\times m=-1$.

  Multiplying by $2$ results in $m=-2$.

  Now, we can write the equation $y=-2x+b$.

  We still have to determine the y-intercept, so we plug the coordinates of $B(7,5)$ into the equation and solve for $b$:

  $\begin{array}{rcl} 5&=&-~~2\times 7+b\\ 5&=&-14+b\\ \color{#669900}{+14}&&\color{#669900}{+14}\\ 19&=&~~~~~b \end{array}$

  Finally, we know the equation of the purple line which is perpendicular to the orange line:

  $y=-2x+19$

• Decide which equations are represented by the given graphs.

  Hints

  Keep in mind:

  • increasing lines have a positive slope
  • decreasing lines have a negative slope

  Remember - rise over run.

  "Rise" means the line goes upwards or downwards while "run" means the line goes to the right.

  You can look at the graphs to find the y-intercepts.

  Which part of the equation represents the y-intercept?

  Solution

  Each equation is written in slope-intercept form: $y=mx+b$.

  • $m$ is the slope
  • $b$ is the y-intercept

  The y-intercept is the intersection point of the line and the y-axis. In order to identify the slope, you will have to determine the change in y and divide it by the change in x.

  Green line

  • y-intercept is $2$
  • rise: $1$ (up), run: $2$, so $m=\frac 12$
  • So the equation is $y=\frac12x+2$

  Orange line

  • y-intercept is $0$
  • rise: $2$ (up), run: $1$, so $m=\frac 21=2$
  • So the equation is $y=2x$

  Light purple line

  • y-intercept is $2$
  • rise: $1$ (up), run: $3$, so $m=\frac 13$
  • So the equation is $y=\frac13x+2$

  Dark purple line

  • y-intercept is $3$
  • rise: $1$ (up), run: $4$, so $m=\frac 14$
  • So the equation is $y=\frac14x+3$

• Identify the lines that are parallel or perpendicular to the orange line.

  Hints

  Parallel lines have the same slope.

  Perpendicular lines meet at a right angle.

  At least one line is neither parallel nor perpendicular to the orange line.

  Solution

  Let's consider the orange line: $y=\frac12x+2$.

  You can identify the slope by calculating rise over run of the line. Count the units you go up or down and divide this number by the number of units you go to the right. The slope is positive if the line is increasing, or rises to the right.

  Parallel lines have the same slope, so the green line is parallel to the red line. Both lines have the slope $m=\frac12$.

  Perpendicular lines meet at a right angle, and the two slopes are always the negative reciprocals of one another, which means the product of the two slopes is always $-1$. The blue line is perpendicular to the red line. It has a slope of $m=-2$.

  $m_1 \times m_2=\frac12 \times -2 = -1$

• Write the equations for the given graphs in different forms.

  Hints

  Simplify the slope as much as possible:

  $\frac22=1$

  • slope-intercept form: $y=mx+b$
  • point-slope-form: $y-y_1=m(x-x_1)$
  • standard form: $Ax+By=C$

  • $m$ is the slope
  • $b$ is the y-intercept
  • $P(x_1,y_1)$ is a point on the line

  For standard form, you have to look at both the x- and y-intercepts: $(0,5)$ and $(7,0)$.

  Solution

  Find the solution on the right side.

  Green line

  We're looking for the equation in slope-intercept form, $y=mx+b$.

  • the y-intercept is $1$
  • the slope is given by the change in height divided by the change in width, $m=\frac11=1$

  The equation is: $y=x+1$.

  Violet line

  We're looking for the equation in point-slope form, $y-y_1=m(x-x_1)$.

  • $x_1=0$ is given as part of the equation, so we use the y-coordinate $y_1=4$
  • the slope is given by the change in height divided by the change in width, $m=\frac18$

  The equation is: $y-4=\frac18(x-0)$.

  Orange line

  We're looking for the standard form, $Ax+By=C$.

  • $B=7$ is already given
  • the y-intercept is $(0,5)$
  • plugging $x=0$ and $y=7$ into the equation, we get $7\times 5=C$, so $C=35$
  • the x-intercept is $(7,0)$
  • plugging $x=7$ and $y=0$ into the equation, we get $A\times 7=35$, dividing by $7$ gives us $A=5$

  The equation is: $5x+7y=35$.