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Chapter 3: Problem 39

Find the \(x\) - and \(y\) -intercepts. Then graph each equation. $$ 5 x+6 y=-10 $$

Short Answer

Expert verified
The x-intercept is \((-2, 0)\) and the y-intercept is \((0, -\frac{5}{3})\).

Step by step solution

01

Find the x-intercept

To find the x-intercept, set y to 0 in the equation and solve for x. The equation is \(5x + 6y = -10\). Plug in \(y = 0\): \[5x + 6(0) = -10\]. Simplify to get \(5x = -10\). Then divide both sides by 5: \(x = -2\). So, the x-intercept is \((-2, 0)\).
02

Find the y-intercept

To find the y-intercept, set x to 0 in the equation and solve for y. Plug in \(x = 0\): \[5(0) + 6y = -10\]. Simplify to get \(6y = -10\). Then divide both sides by 6: \(y = -\frac{10}{6} = -\frac{5}{3}\). So, the y-intercept is \((0, -\frac{5}{3})\).
03

Plot the intercepts

On a coordinate plane, plot the x-intercept at \((-2, 0)\) and the y-intercept at \((0, -\frac{5}{3})\).
04

Draw the line

Draw a straight line through the points \((-2, 0)\) and \((0, -\frac{5}{3})\). This is the graph of the equation \(5x + 6y = -10\).

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Key Concepts

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

x-intercept
To find the x-intercept of a linear equation, you set the value of y to zero and solve for x. This gives you the point where the line crosses the x-axis. For example, in the equation 5x + 6y = -10, replace y with 0 and solve for x.

5x + 6(0) = -10 simplifies to 5x = -10. Divide both sides by 5 to get x = -2. This means the x-intercept is at the point (-2, 0). Remember, the x-intercept will always have the y-coordinate equal to zero because that's the definition of crossing the x-axis.
y-intercept
Finding the y-intercept involves setting the value of x to zero and solving for y. This will locate where the line crosses the y-axis. Using the same equation 5x + 6y = -10, we set x to 0:

5(0) + 6y = -10 simplifies to 6y = -10. When we divide both sides by 6, we get y = -10/6, which further simplifies to y = -5/3. Therefore, the y-intercept is at the point (0, -5/3). The y-intercept always has the x-coordinate equal to zero because it occurs where the line crosses the y-axis.
coordinate plane
The coordinate plane is a two-dimensional surface formed by two perpendicular number lines: the x-axis (horizontal) and the y-axis (vertical). It is used to graph equations and points. Each point on the coordinate plane is defined by an ordered pair (x, y).

When graphing, the first number of the pair represents the horizontal position (left or right), and the second number represents the vertical position (up or down). For the given equation 5x + 6y = -10, you plot the intercepts: the x-intercept (-2,0) and the y-intercept (0,-5/3). These points help you draw the graph of the equation accurately.
linear equation
A linear equation is any equation that graphs to a straight line on the coordinate plane. Its general form is Ax + By = C, where A, B, and C are constants. Linear equations have a constant rate of change, meaning they are the simplest kinds of functions to graph. They can have one or more solutions that essentially form a line.

For instance, the equation 5x + 6y = -10 is linear. By calculating the x-intercept and y-intercept, you gather key points (like (-2,0) and (0,-5/3)) that lie on the line. Connect these points with a straight line to visualize the solution. Linear equations are foundational in algebra and are used to describe relationships between variables.

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Most popular questions from this chapter

Find the \(x\) - and \(y\) -intercepts. Then graph each equation. $$ -\frac{2}{3} y=x $$

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. $$ (7,6) \text { and }(7,-8) $$

Segment PQ has the given coordinates for one endpoint P and for its midpoint M. Find the coordinates of the other endpoint \(Q .\) (Hint: Represent \(Q\) by \((x, y)\) and write two equations using the midpoint formula, one involving \(x\) and the other involving \(y .\) Then solve for \(x\) and \(y .\) $$ P(7,10), M(5,3) $$

Solve each inequality. $$ 2 x+5<9 $$

If a line has slope 0.2, then any line parallel to it has slope _____, and any line perpendicular to it has slope _____.
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