Solving Systems of Equations by Graphing
Basics on the topic Solving Systems of Equations by Graphing
A system of equation is two or more equations with the same variables. To determine the point where the equations meet, the solution to the system, there are several methods: graphing, substitution, and elimination.
This video investigates how to use graphs to solve systems of equations. To determine where lines will meet on the coordinate plane (coordinate grid), manipulate each equation in the system, so it is written in slopeintercept form, y = mx + b. You can do this by isolating the yvalue or the vertical coordinate.
Now using information from the equations written in slopeintercept form, you can graph the lines. Remember b is the yintercept, and m is the slope or the rise over the run. First, mark the yintercept, then using the slope value, you can mark a couple other points on the line and last connect the dots to draw a line. Do this for every equation in the system. The point where the lines intersect is the solution to the system and also the ordered pair that will satisfy all the equations in the solution.
How can a system of equations be useful in the real world? Well imagine you want to accidentally bump into someone, if you knew the equation of their path and the equation of your path… To learn more about this topic, watch the video.
Solve systems of equations to find solutions to problems.
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Transcript Solving Systems of Equations by Graphing
Red Riding Hood and the Wolf are both walking through the forest. To figure out if and where they will meet we can solve a system of equations by graphing.
Let's look at a map of the forest. Red Riding Hood starts here and is walking in this direction. The Wolf starts here and is walking in this direction. It looks like they will meet at some point, but lets take a closer look using math.
When you look at the map, you can see that it looks like a cartesian coordinate system.
Red Riding Hood's path is along the line 2y  4 = x. The Wolf's path is along the line y = 2x  10. As you can see, the equation for the Wolf's path is already in slopeintercept form, so we can graph it easily. Let's transform the equation of Red Riding Hood into slopeintercept form as well.
Transforming Equations into SlopeIntercept Form
Slopeintercept form is y = mx + b. The first step in transforming this equation into slopeintercept form is to add 4 to both sides. Now you have 2y = x + 4. It is almost in slopeintercept form.
You just need to move the 2 in front of the y. You can do this by dividing by 2 on both sides. This reduces to y = 1/2 x + 2. Now the equation is in slopeintercept form.
Solving Equations by Graphing
Let's graph the lines. Red Riding Hood's equation is y = 1/2 x + 2. This means that the slope is 1/2 and the yintercept is at (0, 2).
First, let's plot the yintercept (0, 2). Now, because the slope represents rise over run, you can count up 1 and right 2 to plot the next point. You can keep counting up one and right two to plot more points.
The Wolf's equation is y = 2x  10. This means that the slope is 2 and the yintercept is at (0, 10). Since (0, 10) isn't visible on the graph let's plot our first point somewhere else.
The wolf looks close to where x = 5. When you plug in 5 for x the yvalue is 0. Now you can plot the point (5, 0). Since the slope is 2 you can plot more points by moving up 2 and right one.
The two paths intersect at the point (8, 6). Whether or not Red Riding Hood and the Wolf will meet depends on how fast or slow they are both walking.
Imagine the Wolf's path could be described by y = x/2 + 3. Will they meet? Let's draw the graph.
The yintercept is at (0, 3). Then we plot more points by counting up one and right two. The two lines are now parallel to each other so they will never intersect.
Now imagine the Wolf walked along the line y = 2x/4 + 4/2. Where will they meet? Let's draw the graph. The yintercept is at (0, 2). To plot more points we count up 2 and right 4. As you can see the graphs are the same, which means there are infinitely many possible meeting points.
Let's get back to our first scenario. Red Riding Hood and the Wolf are walking toward the intersection point. Red Riding Hood sees the Wolf and is shocked for a split second. Phew, it's only Jim heading to the same Halloween Party. So they decide to walk together.
Solving Systems of Equations by Graphing exercise

Describe how to graph $y=\frac12x+2$.
HintsYou only need two points to draw a line.
If the slope is given as a fraction $\large m=\frac kl$, you go $k$ steps up for a positive slope, or $k$ steps down for a negative slope, and then $l$ steps to the right.
To solve for the yintercept, plug $x=0$ into the equation $y=\frac12x+2$.
SolutionIf an equation is in slopeintercept form, we can graph the line by following these steps:
 plot the yintercept
 using the slope, plot another point
Let's have a look at the equation describing Red Riding Hood's path, $y=\frac12x+2$:
 The yintercept is at $(0,2)$.
 Starting from this point, because the slope is $m=\frac {1}{2}$, we count up one and two to the right.
 We can draw a line that passes through the yintercept and the new point, $(2,3)$.

Use a graph to show the paths of Red Riding Hood and the Wolf.
HintsFirst, if you know it, draw the yintercept.
You need at least two points to draw a line.
If the slope is given as a fraction, for example $m=\frac23$, to find a point on the line, from the yintercept, we can count up two and three to the right.
If the slope is a negative number, for example $m=2$, to find a point on the line, instead of counting up, we count down two and one to the right.
SolutionRed Riding Hood's path can be described by $y=\frac12x+2$.
 The yintercept is $(0,2)$.
 When we look at the slope $m=\frac12$, it shows us to count up one and two to the right, starting from the yintercept.
 Then, we draw a line that passes through the yintercept and the new point.
 The yintercept is at $(0,10)$. Because this point can't be plotted on the map, we need another point on the line. We substitute $5$ for $x$: $y=2\times 510=1010=0$. So $(5,0)$ is one point on the line.
 A look at the slope $m=2$ tells us that we have to go up two and one to the right, starting from the yintercept.
 Then, we draw a line that passes through the point $(5,0)$ and the new point.

Determine whether or not Red Riding Hood and her grandmother meet.
HintsLook for the yintercepts of the lines.
Which parts of the equations would have to be the same for the lines to be parallel?
SolutionUse what you know of the yintercept and slope to help you solve this problem.
 In two pictures you can see parallel lines. Lines can only be parallel if the slopes are the same. The slope of Red Riding Hood's path is $m=\frac12$, and the slope of her grandma's path is $m=2.5$. So the lines are definitely not parallel. Therefore, you can rule out those two pictures.
 There is one other picture where the two lines have the same yintercept. But the yintercept of Red Rinding Hood's path is $b=2$, and the yintercept of her grandma's path is $b=6$. So this can't be the right picture either.
 You are left with the picture on the right.
 Red Riding Hood's path is given by the equation $y=\frac12x+2$.
 With the yintercept at $(0,2)$ and the slope $m=\frac12$, we can graph the line you can see on the right.
 Grandmother's path is given by the equation $y=2.5x6$.
 The yintercept is at $(0,6)$. Because this point can't be seen on the map, we need another point of the line. If you plug in $4$ for $x$ it gives us $y=2.5\times 46=106=4$, so one point on the line is $(4,4)$.
 The slope is $m=2.5=\frac52$. You can see the graph of the line in the picture on the right.
In this picture we can also find the meeting point $(4,4)$. Let's check it by plugging in $4$ for $x$:
 Red Riding Hood: $y=\frac12\times 4+2=2+2=4$
 Grandmother: $y=2.5\times 46=106=4$ $\surd$

Decide if the lines have a point of intersection.
HintsPlug in $0$ for $x$. This gives you the yintercept.
For example: $m=\frac32$ shows us to count:
 three down
 two to the right.
Draw both lines into a coordinate system to figure out the intersection.
At the intersection point, both equations share the same x and ycoordinates, so you can set the two equations equal and solve for x.
Solution $y=2x3$.
 $y=\frac12x+2$.
In this picture you can also see the point of intersection $(2,1)$.
Let's check this point by plugging in $2$ for $x$:
 $y=2\times 23=43=1$
 $y=\frac12\times 2+2=1+2=1$ $\surd$

Describe how to change $2y4=x$ into slopeintercept form.
HintsThe slopeinercept form is $y=mx+b$.
 $m$ is the slope
 $b$ is the ycoordinate of the yintercept
The xcoordinate of the yintercept is always $0$.
To manipulate the equation use opposite operations:
 The opposite operation of addition is subtraction, and vice versa.
 The opposite operation of multiplication is division, and vice versa.
Whatever you do to one side of the equation, you have to do to the other side.
SolutionTo graph a linear equation, it is helpful to manipulate the equation into slopeintercept form:
$y=mx+b$
 $m$ is the slope
 $b$ is the yintercept
 adding $4$ leads to $2y=x+4$
 dividing by $2$ gives us the slopeinterceptform, $y=\frac12x+2$

Analyze whether or not the lines have a point of intersection with the line of $y=\frac13x+2$
HintsInto one coordinate system, graph each equation in a different color.
The slope represents the steepness of a line.
If two lines have the same slope, the yintercepts define whether they have an infinite amount of common points or none at all.
SolutionThe slope defines if two lines
 have a point of intersection
 are parallel and have no common points or
 are identical and have an infinite amount of common points
Two parallel lines have no common points at all.
Two identical lines have an infinite amount of common points.
Two lines with different slopes may have exactly one point of intersection.
 The line of $y=\frac26x+1=\frac13x+1$ is parallel to the one above, but not identical. The two lines don't have any common points.
 The line of $y=x6$ has a point of intersection with the line above. .
 The line of $y=\frac13x+\frac42=\frac13x+2$ is identical to the one above.
$\begin{array}{rcr} \frac13x+2&=&x6\\ \color{#669900}{x}&&\color{#669900}{x}\\ \frac43x+2&=&6\\ \color{#669900}{2}&&\color{#669900}{2}\\ \frac43x&=&8\\ \color{#669900}{\times 3}&&\color{#669900}{\times 3}\\ 4x&=&24 \end{array}$
Now we divide by $4$ and get
$x=\frac{24}{4}=6$
But this is only the x coordinate of the point of intersection. We get the y coordinate by plugging in $x=6$ into either of the equations:
$y=66=0$
So, the point of intersection is at $S(6,0)$.