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Grade 10

Graphical method

Lesson

So far we've had a look at what simultaneous equations are and at some of the ways to solve them. As with all algebraic expressions, simultaneous equations can also be expressed as graphs on a number plane. In coordinate geometry, we know that each graph represents ALL the possible solutions of a related equation. In other words, if a point is on a graph, it must solve its equation. In simultaneous equations, we mostly deal with linear equations, which can be represented as straight broken line graphs. Our aim is then to find a solution that solves BOTH equations, and graphically this means finding the point of intersection of the two straight lines.

Let's have a look at an example, where we want to find the solution to the simultaneous equations $y=5x$y=5x and $y=x+2$y=x+2. So then we would plot the two equations as graphs. Remember there are two ways to visualise linear equations as graphs: either through finding its intercepts or finding its slope-intercept form. Here I have drawn the $y=5x$y=5x line as red and the $y=x+2$y=x+2 line as green:

We can then see that there is only one intersection point and it is $(0.5,2.5)$(0.5,2.5). Therefore the solution that solves the two equations must be when $x=0.5$x=0.5 and $y=2.5$y=2.5.

 

Parallel lines

Do all pairs of simultaneous linear equations have a solution? Well let's think about this graphically: is it possible to graph two straight lines that never cross over? Of course, it happens when they're parallel! Let's remind ourselves that to find the slope of a linear equation all we have to do is put it in the slope intercept form $y=mx+b$y=mx+b and $m$m will be our slope. This means that for example, the simultaneous equations $y=3x-1$y=3x1 and $y=3x+6$y=3x+6 will never have a solution since they both have a slope of $3$3.

 

Examples

Question 1

Consider the following linear equations:

$y=2x-4$y=2x4 and $y=-2x-4$y=2x4

  1. What are the slope and $y$y-intercept of the line $y=2x-4$y=2x4?

    slope $\editable{}$
    $y$y-value of $y$y-intercept $\editable{}$
  2. What are the intercepts of the line $y=-2x-4$y=2x4?

    $x$x-value of $x$x-intercept $\editable{}$
    $y$y-value of $y$y-intercept $\editable{}$
  3. Plot the lines of the 2 equations on the same graph.

    Loading Graph...

  4. State the values of $x$x and $y$y which satisfy both equations.

    $x$x = $\editable{}$

    $y$y = $\editable{}$

 

question 2

Consider the two equations $3x-y=5$3xy=5 and $2x+y-1=0$2x+y1=0

a) What are the slopes and $y$y-intercepts of the two equations?

Think: The slope-intercept form looks like $y=mx+b$y=mx+b, where $m$m is the slope and $b$b the $y$y-intercept

Do:

$3x-y$3xy $=$= $5$5
$3x-y-5$3xy5 $=$= $0$0
$y$y $=$= $3x-5$3x5

The slope of $3x-y=5$3xy=5 is $3$3 and the $y$y-intercept is $-5$5

$2x+y-1$2x+y1 $=$= $0$0
$2x+y$2x+y $=$= $1$1
$y$y $=$= $-2x+1$2x+1

The slope of $2x+y-1$2x+y1 is $-2$2 and the $y$y-intercept is $1$1

b) Using the slope-intercept form graph the two equations and find the solution that satisfies both

Think: Slope means rise over run, and a solution that satisfies both equations will be the intersection of their graphs

Do:

I've graphed $3x-y=5$3xy=5 as green and $2x+y-1=0$2x+y1=0 as red. The intersection point is $(1.2,-1.4)$(1.2,1.4). Therefore the solution to both equations is $x=1.2$x=1.2 and $y=-1.4$y=1.4

 
question 3

Consider the following linear equations:

$y=5x-7$y=5x7 and $y=-x+5$y=x+5

  1. Plot the lines of the two equations on the same graph.

    Loading Graph...

  2. State the values of $x$x and $y$y which satisfy both equations.

    $x$x = $\editable{}$

    $y$y = $\editable{}$

 

Question 4

Consider the two equations $2x-6+y=0$2x6+y=0 and $15-2y=4x$152y=4x. Is there a solution that satisfies both?

Think: Parallel lines don't have a solution that satisfies both

Do:

Let's put both equations in $y=mx+b$y=mx+b form to find their slopes:

$2x-6+y$2x6+y $=$= $0$0
$-6+y$6+y $=$= $-2x$2x
$y$y $=$= $-2x+6$2x+6
$15-2y$152y $=$= $4x$4x
$-2y$2y $=$= $4x-15$4x15
$y$y $=$= $-2x+\frac{15}{2}$2x+152

The two equations have the same slope of $-2$2, so are parallel. Therefore there are no solutions that satisfy both equations.

 

 

Outcomes

10P.LR3.02

Solve systems of two linear equations involving two variables with integer coefficients, using the algebraic method of substitution or elimination

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