4. Graphing

Graph Linear Equations in Two Variables

4. Graphing

Graph Linear Equations in Two Variables: Videos & Practice Problems

Topic summary

Graphing linear equations in two variables involves plotting ordered pairs that satisfy the equation $a x + b y = c$ . By substituting chosen x-values, students solve for y to find points, then connect these points with a line representing all solutions. This process highlights the relationship between variables and the concept of a polynomial's terms and coefficients. Understanding how to select logical x-values and interpret the graph deepens comprehension of linear functions and their graphical representation in standard form.

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Graphing Linear Equations in Two Variables

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Graphing Linear Equations in Two Variables Video Summary

Graphing linear equations in two variables involves understanding how ordered pair solutions relate to the equation and how these points can be represented visually on a coordinate plane. A linear equation in two variables typically takes the form ax + by = c, where a, b, and c are constants, and x and y are variables. To graph such an equation, the key is to find multiple ordered pairs (x, y) that satisfy the equation and then plot these points on a graph.

To find these ordered pairs, you start by selecting values for x and then solving for the corresponding y values. For example, consider the equation \$2x + y = 1$. If you substitute $x = -1$, the equation becomes $2(-1) + y = 1$, which simplifies to $-2 + y = 1$. Solving for y gives $y = 3$, so the ordered pair is $(-1, 3)$. Repeating this process for other values such as $x = 0$, $x = 1$, and $x = 2$ yields the points $(0, 1)$, $(1, -1)$, and $(2, -3)\( respectively.

Once these points are plotted on the coordinate plane, connecting them with a straight line reveals the graph of the linear equation. This line represents all possible solutions to the equation, not just the points calculated. The line extends infinitely in both directions, indicating that every point on the line is a solution to the equation.

When choosing x values, it is often helpful to select values that are easy to compute and that cover a range of points to clearly define the line. Typically, selecting at least three points ensures accuracy in graphing. If specific x values are not provided, you can choose your own, such as \)-1$, $0$, $1$, and $2$, to get a clear picture of the graph.

This method of graphing linear equations highlights the relationship between algebraic expressions and their geometric representations. By understanding how to find ordered pairs and plot them, you can visualize the solutions to two-variable linear equations effectively. This foundational skill is essential for exploring more complex topics in algebra and coordinate geometry.

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Graphing Linear Equations in Two Variables Example 1

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Graphing Linear Equations in Two Variables Example 1 Video Summary

To graph the linear equation y = 4x − 3, the most effective method is to plot specific points by selecting values for x and calculating the corresponding y values. This approach helps visualize the slope and position of the line on the coordinate plane.

Starting with x = −1, substitute into the equation: $y = 4(-1) - 3 = -4 - 3 = -7$. This point (−1, −7) lies far below the typical graph window, indicating the line will extend steeply downward in that region. Next, for x = 0, calculate $y = 4(0) - 3 = 0 - 3 = -3$, placing the point (0, −3) on the y-axis. This confirms the line crosses the y-axis at −3, known as the y-intercept.

Choosing x = 1 gives $y = 4(1) - 3 = 4 - 3 = 1$, so the point (1, 1) lies above the x-axis. Plotting these points reveals the line’s steep positive slope of 4, meaning for every unit increase in x, y increases by 4 units. This slope is a key characteristic of the line, dictating its angle relative to the x-axis.

By connecting these points with a straight line, the graph of the equation emerges clearly. If initial points fall outside the visible graph area, adjusting the chosen x values to find points within the graph window ensures an accurate sketch. This method of plotting points and connecting them is fundamental for graphing linear equations and understanding their behavior visually.

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Graphing Linear Equations in Two Variables Example 2

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Graphing Linear Equations in Two Variables Example 2 Video Summary

To determine whether an ordered pair is a solution to the linear equation 3x + 4y = 1, you can either substitute the values directly into the equation or use the graph of the equation to check if the point lies on the line. For example, consider the ordered pair (0, 1). When x = 0 and y = 1, the point is located above the line on the graph, indicating it is not a solution. Substituting into the equation confirms this:
$3(0) + 4(1) = 0 + 4 = 4 \neq 1$.

Next, examine the point (-5, 4). On the graph, this point lies exactly on the line, suggesting it is a solution. Substituting into the equation verifies this:
\$3(-5) + 4(4) = -15 + 16 = 1\(.

Finally, consider the point (3, -2). This point also lies on the line in the graph, indicating it satisfies the equation. Substituting the values confirms:
\)3(3) + 4(-2) = 9 - 8 = 1$.

Thus, the ordered pairs (-5, 4) and (3, -2) are solutions to the equation 3x + 4y = 1, while (0, 1) is not. This approach highlights how graphing can provide a quick visual method to identify solutions to linear equations, and substitution serves as a reliable way to verify these solutions algebraically.

Problem

Graph the given linear equation.
$3 x - 2 y = 4$

Graph of the linear equation 3x - 2y = 4 shown as a downward sloping line on an x-y coordinate plane.

Graph of the vertical line x = 3 on a coordinate plane with labeled x and y axes.

Graph of the linear equation 3x - 2y = 4 showing a line passing through points (-2, -5) and (2, 4) on a coordinate plane.

Graph of the linear equation 3x - 2y = 4 shown as an orange line crossing the coordinate plane.

Problem

Graph the given linear equation.
$- 2 x = y - 6$

Graph of the linear equation -2x = y - 6 shown as a downward sloping line on an x-y coordinate plane.

Graph of the linear equation −2x = y − 6 shown as a line passing through points on a coordinate plane.

Graph of the linear equation -2x = y - 6 showing a vertical line at x = 3 on a coordinate plane.

Graph of the linear equation -2x = y - 6 shown as a vertical line at x = 3 on a coordinate plane.

Problem

Graph the given linear equation.
$y = \frac{1}{2} x - 2$

Graph of the linear equation y equals one-half x minus two on a coordinate plane with labeled axes.

Graph showing a straight line with positive slope passing through points (-4,-4) and (4,2) on a coordinate plane.

Graph of the linear equation y equals one-half x minus two on a coordinate plane with labeled axes.

Graph of the linear equation y equals one-half x minus two, showing a line passing through points (0, -2) and (4, 0).

Here’s what students ask on this topic:

To graph a linear equation in two variables, such as $a x + b y = c$ , you first find ordered pairs (x, y) that satisfy the equation. Start by choosing values for $x$ , then substitute these into the equation to solve for $y$ . Each solution gives you a point to plot on the coordinate plane. After plotting at least three points, connect them with a straight line. This line represents all possible solutions to the equation. The process helps visualize the relationship between $x$ and $y$ and shows the linear function graphically.

The standard form of a linear equation in two variables is written as $a x + b y = c$ , where $a$ , $b$ , and $c$ are constants, and $x$ and $y$ are variables. This form is useful because it clearly shows the coefficients of each variable and the constant term. It is the starting point for graphing linear equations by finding ordered pairs that satisfy the equation and plotting them on a coordinate plane.

Plotting at least three points when graphing a linear equation ensures accuracy. While two points are enough to define a line, a third point helps confirm that the points lie on the same line and that no mistakes were made in calculations. This is especially helpful when plotting by hand or when the equation is complex. Once the points are plotted, connecting them with a straight line shows all solutions to the equation.

To find the $y$ -value for a given $x$ -value in a linear equation like $a x + b y = c$ , substitute the chosen $x$ into the equation. Then solve for $y$ by isolating it on one side. For example, if the equation is $2x + y = 1$ and $x = -1$ , substitute to get $2(-1) + y = 1$ . Simplify to $-2 + y = 1$ , then add 2 to both sides to find $y = 3$ . This ordered pair (-1, 3) can then be plotted on the graph.

The graph of a linear equation in two variables represents all the ordered pairs (x, y) that satisfy the equation. When these points are plotted and connected, they form a straight line. This line shows the relationship between the variables and all possible solutions to the equation. The line extends infinitely in both directions, indicating that there are infinitely many solutions, not just the points plotted.