Graphing Linear Equations in Desmos

Finding a Missing y-Value on a Line

Desmos draws the rental line by working out the cost for every number of hours and placing a point for each pair. Here are five of those pairs for $C(h) = 12h + 15$:

Each row is one point on the line: the hours say how far right it sits, and the cost says how high. The line is all of these points together, so when a question gives a point’s x-coordinate, its y-coordinate is the output the equation gives at that $x$. The walkthrough’s question gave 4 hours, so its answer was the cost in the row where $h$ is 4.

Many questions describe a line with an equation written with $y$, such as $y = 6x - 7$, and give a point with its y-value missing. In Desmos, an equation like that draws the line but has no name, so there is no way to ask for $y$ at one chosen $x$. Writing the same equation as a function with a name, $f(x) = 6x - 7$, fixes that.

In the simulator, $y=6x-7$ drew the line and showed no number. Typing $f\left(x\right)=6x-7$ gave the same equation the name $f$, and $f\left(5\right)$ showed 23, so the point on the line where $x$ is 5 is $(5, 23)$, and $b$ is 23.

FastestFastest method: Algebra For a small whole-number input, working out the equation in your head is quicker than typing: for $y = 6x - 7$ at $x = 5$, 6 times 5 is 30, and 30 minus 7 is 23. Desmos saves time when the input is negative or a decimal, where a sign or a decimal place is easy to lose by hand. Typing the function’s name with the input, such as $f\left(-2.5\right)$, makes Desmos treat the whole negative number as the input, as the lesson on Desmos typing tricks explains.

Known Outputs and Unknown Inputs

The rental model, $C(h) = 12h + 15$, can also be asked about the other way around. “For how many hours can a customer rent a bike for a total cost of 87 dollars?” gives the output, 87 dollars, and asks for the input that produces it. That question is the equation $C(h) = 87$, and typing $C\left(87\right)$ would answer a different question: the cost of renting a bike for 87 hours.

When a question gives the output, the answer is an input, and a horizontal line at the height of that output, such as $y = 87$, crosses the graph exactly where that input is. The equation $12h + 15 = 87$ also takes only two short steps by hand, so the rental question can be solved either way.

In the simulator, $C\left(87\right)$ showed 1,059, the cost of an 87-hour rental, which is choice D. The horizontal line $y=87$ has a height of 87 at every point, so it meets the rental line only at the point whose cost is 87 dollars, $(6, 87)$, and the hours are that point’s first number, 6. By hand, subtracting the 15-dollar fee from 87 leaves 72, and dividing 72 by the hourly rate, 12, gives the same 6 hours.

The Y-Intercept and the X-Intercept

Two points on a slanted line involve the number 0. The y-intercept is where the line crosses the y-axis, and every point on that axis has an x-value of 0, so the y-intercept is the point whose input is 0. The x-intercept is where the line crosses the x-axis, where every point has a y-value of 0, so it is the point whose output is 0. In a model, the output at an input of 0 is the starting value, the amount before any change happens.

In the simulator, $B\left(t\right)=96-8t$ met the vertical axis at $(0, 96)$: at 0 hours, the moment of unplugging, the charge is 96 percent, the starting value. It met the horizontal axis at $(12, 0)$: after 12 hours, the charge is 0. The y-intercept answers a question that sets the input to 0, and the x-intercept answers one that sets the output to 0, so they are two different points with two different answers. Asking when the battery runs out sets the charge to 0, which is why the answer was 12 hours and 96 was a wrong choice.

FastestFastest method: Read it from the equation When a line is written as $y = mx + b$, or a model as a starting amount plus or minus a rate times the input, as in $B(t) = 96 - 8t$, the number standing alone is the y-value of the y-intercept, so there is nothing to graph. A number inside parentheses is different: $y = 4(x + 3)$ crosses the y-axis at 12, not 3, because 4 times 3 is 12. The x-intercept always takes a short solve, such as $96 - 8t = 0$, or one click in Desmos.

Equations in Standard Form

Some linear equations are printed with $x$ and $y$ on the same side, such as $3x + 4y = 48$. That arrangement, $Ax + By = C$, is called standard form. Desmos draws an equation like this exactly as it is printed, so you can type it without solving it for $y$ first.

In the simulator, $3x+4y=48$ drew a line with gray points at $(16, 0)$ on the x-axis and $(0, 12)$ on the y-axis. The two gray points on the axes are the equation’s intercepts, and each one is what the equation gives when the other letter is 0. By hand, each intercept takes one division:

An equation in this form often describes two amounts that share a total. If adult tickets cost 3 dollars, child tickets cost 4 dollars, and a group spends 48 dollars on $x$ adult tickets and $y$ child tickets, then $3x + 4y = 48$. The x-intercept, $(16, 0)$, means 16 adult tickets and no child tickets, and the y-intercept, $(0, 12)$, means 12 child tickets and no adult tickets.

Slope From Two Points

The slope of a line is how much its output changes when the input goes up by 1. In a model, the slope is a rate of change: in $B(t) = 96 - 8t$, the charge drops by 8 percentage points every hour, so the slope is $-8$. Between any two points on a line, the slope is the change in $y$ divided by the change in $x$, which the battery line’s two intercepts, $(0, 96)$ and $(12, 0)$, confirm:

FastestFastest method: Read it from the equation When $y$ is alone, as in $y = mx + b$, the slope is the number multiplying $x$, so reading it beats graphing. An equation with $x$ and $y$ on the same side hides its slope, and there two clicked points can find it. The lesson on graphing linear equations by hand covers slope-intercept form and solving for $y$.

In the simulator, $2x-5y=20$ showed gray points at $(0, -4)$ and $(10, 0)$. From the first point to the second, $y$ rises 4 while $x$ rises 10:

Any two points on a line give its slope, so the two gray intercepts give the slope that an equation with $x$ and $y$ on one side hides. Take both changes in the same direction: going from $(10, 0)$ back to $(0, -4)$ gives $\frac{-4}{-10}$, the same $\frac{2}{5}$.

Graphing Each Side of an Equation

In the rental question, one side of $12h + 15 = 87$ was a single number, so a horizontal line at a height of 87 found the answer. The equation $3(x + 2) = 5x - 8$ has $x$ on both sides, so neither side stays at one height. Each side can still be drawn as a line of its own: typing $y=3\left(x+2\right)$ graphs the value of the left side for every $x$, and $y=5x-8$ graphs the value of the right side.

In the simulator, $y=3\left(x+2\right)$ and $y=5x-8$ crossed at $(7, 27)$. At a crossing, both sides give the same output for the same input, so the crossing’s x-value is the solution and its y-value is the value both sides share. Putting 7 into each side of $3(x + 2) = 5x - 8$ confirms it:

FastestFastest method: Desmos By hand, $3(x + 2) = 5x - 8$ takes four steps: distribute the 3, subtract $3x$, add 8, and divide by 2, and each one is a chance for a sign slip. Two typed lines and one click reach the same crossing, and substituting checks it. A question that uses both numbers of a crossing, such as a system of two equations in $x$ and $y$, is the subject of the lesson on graphing systems of equations in Desmos.

No Solution and Infinitely Many Solutions

Graphing each side also shows when an equation has no solution, or when every number is a solution. Both happen when the two sides graph as lines with the same slope.

In the simulator, $y=2\left(3x-1\right)$ and $y=6x+5$ drew two lines that never meet. Distributing shows why: the left side is $6x - 2$, so both sides have slope 6, and their y-intercepts, $-2$ and 5, differ. The right side is always 7 more than the left, so no value of $x$ makes them equal. Parallel lines mean the equation has no solution, and one line drawn exactly on top of the other means every value of $x$ is a solution.

The second case happens when the two sides are the same expression written two ways. In $4(x - 2) + 3 = 4x - 5$, the left side distributes to $4x - 8 + 3$, which is $4x - 5$, exactly the right side. Desmos draws the second line on top of the first, so you see one line and no gray crossing point, which can look like no solution at all.

FastestFastest method: Read it from the equation Once each side is simplified to a number times $x$ plus a number, compare them: the same x-term with different constants means no solution, and the same x-term with the same constant means infinitely many solutions. Graphing first helps when simplifying would take several steps. If the graph then shows a single line with no gray crossing point, simplify both sides to confirm that the two lines lie on top of each other.

Summary: Given and Asked Numbers on a Line

Practice: Linear Equations in Desmos

These questions mix every kind of linear question in the lesson, from easiest to hardest, so the first step on each one is deciding what it gives you: an input, an output, a line to describe, or an equation with two sides.