Solving System of Equations Graphically

An alternative method for determining the number of solutions to a system of equations is to compare the slopes and y-intercepts of the equations. y= mx+ b We can use the slope-intercept form of each equation, where m is the slope and the point (0, b) is the y-intercept, to help us with this. There are three possibilities when comparing two linear equations in a system.

Let's rewrite the equations in the given system in slope-intercept form, highlighting the m and b values.

From the table, we can see that the slopes of the lines are equal, so the lines are either parallel or are the same line. Looking at the y-intercepts, we can tell the lines are the same because the point at which each line crosses the y-axis is the same. Therefore, the system has infinitely many solutions.

Let's start by writing a system of equations for the situation. Let t be the number of text messages Zain sends or receives and c be the total cost of the plan. t &= number of text messages c &= cost of the plan For the Chit Chat plan, Zain has to pay $ 30 per month plus $ 0.25 for each text message sent or received, so 0.25 t. Let's use this information to write an equation for the total cost of this plan. Chit Chat c= 30+ 0.25 t The second plan, Textopia, costs $ 55 per month, regardless of the number of messages sent or received. Let's write an equation for this plan. Textopia c= 55 Together these equations form a system of equations. c= 30+ 0.25 t c= 55 Let's graph the system to find its solution. We can graph the first equation by plotting the y-intercept at (0,30) and then using the slope of 0.25 to move 1* 40=40 units right and 0.25*40=10 units up. We can graph the second equation as a horizontal line intersecting the y-axis at (0,55).

Now we can identify the coordinates of the point of intersection of the lines.

We can see that the graphs intersect at the point (100,55). Since t represents the number of text messages, this tells us that Zain would have to send or receive 100 text messages for the plans to cost the same each month.

We can decide which plan is better for sending or receiving only 60 messages per month by analyzing the graph we created in Part A. Since the horizontal line is constant at y=55, we only need to analyze the graph of the first plan Chit Chat.

We can see that for the first cell phone plan, Zain would only pay about $45 per month, which is less than the $55 for the Textopia plan. Therefore, it is more logical for Zain to buy the first plan.

We are given two equations that we can use to form a system of equations. 3x+5y=24 12x+20y=82 Let's graph the system to find its solution. First, we need to rewrite the equations from standard form into slope-intercept form. Standard Form Ax+By=C ⇓ Slope-Intercept Form y=mx+b Let's do it!

Next, we graph the first equation by plotting the y-intercept at (0,4.8) and then using the slope of - 0.6 to move 1* 3=3 units right and 0.6* 3=1.8 units down. We will then draw a line through the two points. After that, we will follow the same procedure to graph the second line.

We can see that the lines do not intersect and are parallel to each other. This means that the system of equations has no solution. Ali must have made a mistake in his clues!

We know that Dominika sold a total of 32 candles. If we let s be the number of small candles sold and l be the number of large candles sold, we can write an equation to represent the total number of candles sold. s+ l=32 Let's rearrange this equation to isolate the l-variable to make it easier to use this equation later. l=- s+32 We also know that a large candle sells for $8 and a small candle for $6. Let's use this information to express how much money Dominika collected in terms of s and l.

Since Dominika collected $ 228, let's write an equation to relate the total amount of money earned with the number of the candles sold. 8 l+ 6 s=228 Now we will rearrange this equation by isolating l since it will help us graph the equation. In other words, we will rewrite the equation in slope-intercept form.

Now that we have both equations in slope-intercept form, we can plot them using their y-intercepts and slopes and then find the point of intersection.

From the graph, we can see that the solution seems to be (14,18). We can also check this solution by substituting it into the equations. Let's start with the first one.

Now let's check the second one.

Since both equations resulted in a true statment, the point (14,18) is the solution to the system. This means that Dominika sold 14 small candles and 18 large candles.