Graph x-y=6 4x+y=4
This problem is asking to graph two linear equations on the same coordinate plane. Each linear equation represents a straight line, with the first line defined by the equation x - y = 6, and the second line defined by the equation 4x + y = 4. The goal is to plot these lines on a graph, illustrating both their individual positions and their relationship to each other, such as whether they intersect, are parallel, or are the same line. The graphing process will include finding points that satisfy each equation and then connecting these points to form the lines on the graph.
$x - y = 6$$4 x + y = 4$
Answer
Solution:
Step:1
Plot the line for the equation $x - y = 6$.
Step:1.1
Isolate $y$ in the equation.
Step:1.1.1
Move $x$ to the right side of the equation to get $-y = 6 - x$.
Step:1.1.2
Multiply every term in the equation $-y = 6 - x$ by $-1$ to make $y$ positive.
Step:1.1.2.1
Apply the multiplication by $-1$ to each term: $\frac{-y}{-1} = \frac{6}{-1} - \frac{x}{-1}$.
Step:1.1.2.2
Convert the left side to a positive $y$.
Step:1.1.2.2.1
When two negatives are divided, the result is positive: $y = \frac{6}{-1} - \frac{x}{-1}$.
Step:1.1.2.2.2
Simplify $y$ divided by $1$: $y = -6 + x$.
Step:1.1.2.3
Simplify the right side of the equation.
Step:1.1.2.3.1
Break down each term.
Step:1.1.2.3.1.1
Calculate $6$ divided by $-1$: $y = -6 + x$.
Step:1.1.2.3.1.2
A negative divided by a negative is positive: $y = -6 + x$.
Step:1.1.2.3.1.3
Simplify $x$ divided by $1$: $y = -6 + x$.
Step:1.2
Express the equation in the form $y = mx + b$.
Step:1.2.1
Identify the slope-intercept form as $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.
Step:1.2.2
Switch the order of $-6$ and $x$: $y = x - 6$.
Step:1.3
Determine the slope and y-intercept from the slope-intercept form.
Step:1.3.1
Extract the values of $m$ and $b$ from $y = mx + b$: $m = 1$, $b = -6$.
Step:1.3.2
The line's slope is $m$, and the y-intercept is at $b$: Slope: $1$, y-intercept: $(0, -6)$.
Step:1.4
To graph the line, choose two $x$ values and calculate the corresponding $y$ values.
Step:1.4.1
Reaffirm the order $-6$ and $x$: $y = x - 6$.
Step:1.4.2
Tabulate the $x$ and $y$ values: $\begin{array}{c|c} x & y \\ \hline 0 & -6 \\ 1 & -5 \end{array}$
Step:1.5
Draw the line using the slope and y-intercept, or the plotted points.
Step:2
Plot the line for the equation $4x + y = 4$.
Step:2.1
Isolate $y$ by subtracting $4x$ from both sides: $y = 4 - 4x$.
Step:2.2
Express the equation in slope-intercept form.
Step:2.2.1
Recognize the slope-intercept form as $y = mx + b$.
Step:2.2.2
Rearrange the terms $4$ and $-4x$: $y = -4x + 4$.
Step:2.3
Determine the slope and y-intercept from the slope-intercept form.
Step:2.3.1
Extract the values of $m$ and $b$ from $y = mx + b$: $m = -4$, $b = 4$.
Step:2.3.2
The line's slope is $m$, and the y-intercept is at $b$: Slope: $-4$, y-intercept: $(0, 4)$.
Step:2.4
To graph the line, choose two $x$ values and calculate the corresponding $y$ values.
Step:2.4.1
Reaffirm the order $4$ and $-4x$: $y = -4x + 4$.
Step:2.4.2
Tabulate the $x$ and $y$ values: $\begin{array}{c|c} x & y \\ \hline 0 & 4 \\ 1 & 0 \end{array}$
Step:2.5
Draw the line using the slope and y-intercept, or the plotted points.
Step:3
Overlay the two graphs on the same coordinate system to visualize the lines represented by $x - y = 6$ and $4x + y = 4$.
Step:4
Analyze the intersection point of the two lines, if any, to find the solution to the system of equations.
Knowledge Notes:
To solve a system of linear equations graphically, one must plot each equation on the same coordinate system and identify the point of intersection, which represents the solution to the system. The steps involve:
Rearranging each equation into slope-intercept form, $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.
Identifying the slope and y-intercept for each line.
Selecting two or more values for $x$ to find corresponding $y$ values to plot points on the graph.
Drawing each line on the graph using the points or the slope and y-intercept.
Finding the intersection of the lines, which gives the solution to the system.
When dividing or multiplying by negative numbers, the sign of the result will change. For example, dividing a negative by a negative results in a positive, as seen in the steps where $-y$ is converted to $y$.
The slope-intercept form is useful for graphing because it easily shows the slope of the line and where it crosses the y-axis. The slope indicates the steepness and direction of the line, while the y-intercept indicates the point where the line crosses the y-axis.
