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Chapter 5: Problem 3

For the following formulae, find \(y\) at the given values of \(x\) : (a) \(y=3 x+2, x=-1, x=0, x=1\) (b) \(y=-4 x+7, x=-2, x=0, x=1\)

Short Answer

Expert verified
For (a): x = -1, y = -1; x = 0, y = 2; x = 1, y = 5. For (b): x = -2, y = 15; x = 0, y = 7; x = 1, y = 3.

Step by step solution

01

Substitute and Calculate for (a) when x = -1

Substitute \(x = -1\) into the equation \(y = 3x + 2\):\[y = 3(-1) + 2\]Calculate:\[y = -3 + 2 = -1\]Thus, when \(x = -1\), \(y = -1\).
02

Substitute and Calculate for (a) when x = 0

Substitute \(x = 0\) into the equation \(y = 3x + 2\):\[y = 3(0) + 2\]Calculate:\[y = 0 + 2 = 2\]Thus, when \(x = 0\), \(y = 2\).
03

Substitute and Calculate for (a) when x = 1

Substitute \(x = 1\) into the equation \(y = 3x + 2\):\[y = 3(1) + 2\]Calculate:\[y = 3 + 2 = 5\]Thus, when \(x = 1\), \(y = 5\).
04

Substitute and Calculate for (b) when x = -2

Substitute \(x = -2\) into the equation \(y = -4x + 7\):\[y = -4(-2) + 7\]Calculate:\[y = 8 + 7 = 15\]Thus, when \(x = -2\), \(y = 15\).
05

Substitute and Calculate for (b) when x = 0

Substitute \(x = 0\) into the equation \(y = -4x + 7\):\[y = -4(0) + 7\]Calculate:\[y = 0 + 7 = 7\]Thus, when \(x = 0\), \(y = 7\).
06

Substitute and Calculate for (b) when x = 1

Substitute \(x = 1\) into the equation \(y = -4x + 7\):\[y = -4(1) + 7\]Calculate:\[y = -4 + 7 = 3\]Thus, when \(x = 1\), \(y = 3\).

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Key Concepts

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

Finding y Values
Finding values of \( y \) in linear equations is an essential skill in algebra. Linear equations often appear in the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. To find \( y \), you need to substitute a specific value of \( x \) into the equation.

Here鈥檚 a simple way to approach this:
  • Identify the equation and the given value of \( x \).
  • Substitute the value of \( x \) into the equation.
  • Simplify the expression to calculate the corresponding \( y \) value.
Let's look at an example: for the equation \( y = 3x + 2 \) and \( x = -1 \), substitute \( -1 \) for \( x \) to get \( y = 3(-1) + 2 = -1 \). Repeat this process for each value of \( x \) you need to check.
Substitution Method
The substitution method is a straightforward technique used to find unknown variable values. It involves placing the known value of a variable into an equation to find the unknown one. In these exercises, we substitute \( x \) values into the linear equation to calculate \( y \).

Here's how this method works:
  • Take the value of \( x \) provided, for instance, \( x = 0 \).
  • Insert this value into the equation, such as \( y = 3(0) + 2 \).
  • Solve the equation to get the result: \( y = 2 \).
Using this method, you systematically find \( y \) for each \( x \), which helps in understanding how changes in \( x \) impact \( y \). It's a crucial step-by-step process that builds confidence in algebra and problem-solving.
Step-by-Step Calculation
Breaking down the calculation into small, manageable steps is key to solving equations accurately. By approaching equations step-by-step, you ensure each part of the calculation is understood. This method highlights attention to detail and simplifies complex problems.

Here鈥檚 how the process unfolds:
  • **Step 1:** Substitute the \( x \) value into the equation.
  • **Step 2:** Perform the multiplication, i.e., \( 3 \times x \) or \( -4 \times x \).
  • **Step 3:** Add or subtract the constant to complete the calculation.
For example, substituting \( x = 1 \) in \( y = -4x + 7 \), the steps are:
\( y = -4(1) + 7 = -4 + 7 = 3 \).
By consistently using these steps, you develop a strong foundation in solving linear equations efficiently. Regular practice with step-by-step calculations will improve your confidence and precision in math.

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Most popular questions from this chapter

Evaluate without using a calculator: (a) \(\left(\frac{2}{3}\right)^{2}\) (b) \(\left(\frac{2}{5}\right)^{3}\) (c) \(\left(\frac{1}{2}\right)^{2}\) (d) \(\left(\frac{1}{2}\right)^{3}\) (e) \(0.1^{3}\)

For the following formulae, find \(y\) at the given values of \(x\) : (a) \(y=2-x, x=-3, x=-1, x=1, x=2\) (b) \(y=x^{2}, x=-2, x=-1, x=0, x=1, x=2\)

If \(y=\sqrt{x / z}\) find \(y\) if \(x=13.2\) and \(z=15.6\).

If \(\eta=\frac{4 Q_{\mathrm{P}}}{\pi d^{2} L n}\) evaluate \(\eta\) when \(Q_{\mathrm{P}}=0.0003\), \(d=0.05, L=0.1\) and \(n=2\).

If \(P=\frac{3}{Q R}\) find \(P\) if \(Q=15\) and \(R=0.3\).
See all solutions

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