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Chapter 3: Problem 42

Find the \(x\) - and \(y\) -intercepts. Then graph each equation. $$ \frac{5}{7} x+\frac{6}{7} y=-2 $$

Short Answer

Expert verified
The x-intercept is \( ( -\frac{14}{5}, 0 )\) and the y-intercept is \( ( 0, -\frac{7}{3} )\).

Step by step solution

01

- Find the x-intercept

To find the x-intercept, set \( y = 0\) and solve the equation for \( x\).Given equation: \( \frac{5}{7} x + \frac{6}{7} y = -2\)Set \( y = 0\):\( \frac{5}{7} x + \frac{6}{7} \cdot 0 = -2\)So,\( \frac{5}{7} x = -2\)Multiply both sides by \(7\):\( 5x = -14\)Divide by \(5\):\( x = -\frac{14}{5}\)Therefore, the x-intercept is \( ( -\frac{14}{5}, 0 )\)
02

- Find the y-intercept

To find the y-intercept, set \( x = 0\) and solve the equation for \( y\).Given equation: \( \frac{5}{7} x + \frac{6}{7} y = -2\)Set \( x = 0\):\( \frac{5}{7} \cdot 0 + \frac{6}{7} y = -2\)So,\( \frac{6}{7} y = -2\)Multiply both sides by \(7\):\( 6y = -14\)Divide by \(6\):\( y = -\frac{14}{6}\)Therefore, the y-intercept is \( ( 0, -\frac{7}{3} )\)
03

- Graph the equation

To graph the equation, plot the x-intercept and the y-intercept found in the previous steps, then draw a line through these points.The x-intercept is \( (-\frac{14}{5}, 0)\).The y-intercept is \( (0, -\frac{7}{3})\).Mark these points on a coordinate plane and then draw a straight line passing through them to represent the equation.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

graphing linear equations
Linear equations represent straight lines when graphed on a coordinate plane. To graph a linear equation, it's crucial to identify points that lie on the line. An effective method for graphing involves determining the x-intercept and y-intercept of the equation.
These intercepts provide two anchor points, making it easier to illustrate the equation.
For example, to graph the equation \(\frac{5}{7}x + \frac{6}{7}y = -2\), first, find the x- and y-intercepts.
Once identified, plot these points and simply draw a straight line through them.
This method guarantees that the line precisely represents the equation.
coordinate plane
A coordinate plane is a two-dimensional surface where we plot points, lines, and curves.
It consists of two perpendicular axes: the horizontal x-axis and the vertical y-axis.
The point where these axes intersect is called the origin, denoted as \(0,0\).
When graphing a linear equation, it's essential to understand this coordinate system.
For instance, the x-intercept is where the line crosses the x-axis, meaning y equals zero at this point.
Conversely, the y-intercept is where the line crosses the y-axis, implying x is zero there.
By plotting these intercepts (\(-\frac{14}{5},0\) and \(0,-\frac{7}{3}\)), we use the coordinate plane's structure to accurately draw the line.
solving linear equations
Solving linear equations involves finding the values of x and y that satisfy the equation.
The key is simplifying the equation step-by-step to isolate the variables. For our example equation \( \frac{5}{7}x + \frac{6}{7}y = -2\):
  • To find the x-intercept, set y to zero and solve for x.
    \(\frac{5}{7}x = -2\) simplifies to \ x = -\frac{14}{5} \.
  • To find the y-intercept, set x to zero and solve for y.
    \(\frac{6}{7}y = -2\) simplifies to \ y = -\frac{7}{3} \.
These steps clarify the x- and y-values where the line crosses the axes.
Rearranging terms, multiplying, and dividing enables these solutions, making plotting possible.

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Most popular questions from this chapter

Find an equation of the line passing through the given points. (a) Write the equation in standard form. (b) Write the equation in slope-intercept form if possible. $$ (7,6) \text { and }(7,-8) $$

Find the slope of the line through each pair of points.\(\left(\text {Hint:} \frac{\frac{a}{b}}{\frac{c}{d}}=\frac{a}{b} \div \frac{c}{d}\right)\). $$ \left(-\frac{4}{5}, \frac{9}{10}\right) \text { and }\left(-\frac{3}{10}, \frac{1}{5}\right) $$

Find the midpoint of each segment with the given endpoints. $$ (-9,3) \text { and }(9,8) $$

Find an equation of the line passing through the given points. (a) Write the equation in standard form. (b) Write the equation in slope-intercept form if possible. $$ (-2,2) \text { and }(4,2) $$

Find the \(x\) - and \(y\) -intercepts. Then graph each equation. $$ 4 y=3 x $$
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