Problem 92 - Algebra 1

Question

Find the \(x\) -intercept and the \(y\) -intercept of the graph of the equation. $$y=3 x+9$$

Step-by-Step Solution

Answer

The \(x\)-intercept is -3 and the \(y\)-intercept is 9.

Step 1: Identify the y-intercept

We can directly obtain the \(y\)-intercept from the equation. It is the constant term \(c\) in the equation \(y = mx + c\). So, in our equation \(y = 3x + 9\), the \(y\)-intercept is 9.

Step 2: Find the x-intercept

The \(x\)-intercept is the value of \(x\) when \(y = 0\). We can find it by substituting \(y = 0\) in the given equation and solving for \(x\):\[\begin{{align*}}0 &= 3x + 9 \3x &= -9 \x &= -3 .\end{{align*}}\]So, the \(x\)-intercept is -3.

Key Concepts

Intercepts

Intercepts are key elements when working with linear equations. They represent the points where a graph crosses the axes on a coordinate plane.To find the **y-intercept**, look at the linear equation in slope-intercept form: \[y = mx + c\] Here, the y-intercept is the constant term, often labeled as \(c\). This means the graph crosses the y-axis at point \((0, c)\). In the given equation, \(y = 3x + 9\), the y-intercept is 9.
To find the **x-intercept**, you need the graph's point crossing the x-axis, where \(y\) equals zero. By setting \(y = 0\) and solving for \(x\), determine this intercept. For example, with \(y = 3x + 9\), set \(0 = 3x + 9\):

• Subtract 9 from both sides resulting in \(-9 = 3x\)
• Divide by 3 giving \(x = -3\)

Thus, the x-intercept is \(-3\."\), the x-intercept is \(-3\). Intercepts help us understand intercepts and the general direction of a line.

Graphing Linear Equations

Graphing linear equations helps visualize their relationships. It allows you to see where the line crosses the axes, at the intercepts previously discussed.

Begin by identifying points like intercepts, then use these to draw the line. For the equation \(y = 3x + 9\):

• First, plot the y-intercept (0, 9).
• Next, find and plot the x-intercept (-3, 0).

These intercepts are key plotting points. Simply connect them with a straight line. This shows the linear relationship making it easy to see trends or slopes.
Remember, a linear equation forms a straight line. The rapidity of the slope indicates how fast \(y\) values change concerning \(x\). With practice, knowing the equation and plotting becomes more intuitive.

Solving Equations

Solving equations involves finding values of variables that satisfy the equation. With linear equations like \(y = 3x + 9\), you often aim to find intercepts or other specific points.Solving for the x-intercept is straightforward, as seen before:

• Set \(y = 0\): \(0 = 3x + 9\)
• Subtract 9 from each side to isolate constants: \(-9 = 3x\)
• Divide each term by 3 to solve for x: \(x = -3\)

This method confirms values at which the line crosses the x-axis. The solution shows the simplicity of linear equations where each step logically follows from the last.
Sometimes, solving involves rearranging the equation to find a variable's expression, known as solving for \(x\) or \(y\). This practice becomes fundamental in further studies, where solving one equation often helps in solving more complex systems of equations.